Chiefly used in British and Australian English, the phrase ginger group designates a group within a political party or other organisation, which presses for stronger or more radical policy or action.

This phrase occurs, for example, in Rees-Mogg: I’m quitting as ERG chief but won’t have wings clipped, by Kate Proctor, Political Reporter, published in the Evening Standard (London, England) of Thursday 1^st August 2019:

TORY Right-winger Jacob Rees-Mogg has announced he is to quit as chairman of the Brexiteer MPs’ European Research Group.
[…]
The MP for North East Somerset said he had to leave to allow the group to be as effective as possible in holding the Government to its Brexit promises. “I think it is better for a ginger [activist] group to be chaired by somebody who is not a member of the Government. I think people might worry I’d gone too native to run the ERG as effectively as a backbencher could,” he said.

In the phrase ginger group, the noun ginger is used figuratively in the sense of spirit, pep, energy, and the image is of providing stimulus in a party or organisation—as exemplified, below, by the quotation from John Bull (London, England) of Saturday 5^th February 1916, and by the quotation from The Bendigo Advertiser (Bendigo, Victoria, Australia) of Friday 5^th May 1916.

The earliest occurrences of the phrase ginger group that I have found are as follows, in chronological order:

1-: From John Bull (London, England) of Saturday 5^th February 1916:

The “Ginger” Group.
Those Radicals who have joined together to put “ginger” into the Government are described as the “Ginger” group. The fact that the hair of our once pertinacious friend, Handel Booth, is of the Rufus ¹ hue adds colour to the description. Nothing like ginger for pluck!

¹ Colloquially used as a nickname for a red-haired person, the noun rufus is from the classical-Latin adjective rūfus, meaning red-haired, which was frequently used as a Roman surname.

2-: From the Sheffield Daily Telegraph (Sheffield, Yorkshire, England) of Monday 14^th February 1916:

Parliament to-morrow begins the extra session which it recently allowed itself by special Act. The King being unable to come to the House of Lords, there will be no procession in Whitehall, and small pomp inside the Houses. The Commons will assemble at their ordinary time, and be summoned to the House of Lords to hear the Speech from the Throne read in Victorian fashion by the Lord Chancellor sitting, with four other Royal Commissioners. After a short interval both Houses will debate the Address […].
In the Commons the Unionists and Radical “ginger” groups—those back benchers who want to organise effective criticism and keep an eye on the Government—have both arranged to meet before the debate on the Address begins, and to arrange speakers and amendments. It is likely, however, that before any amendment is taken the House will spend some hours in a general debate on the Address.

3-: From an account of the debates in the House of Commons, published in The Birmingham Daily Mail (Birmingham, Warwickshire, England) of Tuesday 15^th February 1916:

The Speaker read the King’s Speech.
[…]
Mr. Stuart Wortley, in the absence of Mr. Henry Chaplin, through illness, continuing the discussion, said they had at the moment no opposition and no parties. There were only groups in the House, some of them anxious to render all assistance possible to the Government, and others of them whose only purpose seemed to be to make it difficult for the Government. (Laughter.) He believed they were called “ginger” groups. The matters dealt with in the Speech were matters of general consent and universal approval. (Cheers.) He hoped that the Government’s programme would include vigorous proposals for national economy.

The earliest occurrence of the phrase ginger group that I have found in an Australian publication is from The Bendigo Advertiser (Bendigo, Victoria) of Friday 5^th May 1916:

OUR LONDON LETTER.
(From Our Special Correspondent).

London, 13^th March.
Rejected at Mile End, Mr. Pemberton-Billing has been more kindly treated by the people of East Herts. They have sent him to the House of Commons with a mandate to put “ginger” into the Government, and especially to insist that the country’s defences against hostile aircraft should be strengthened. […] A great many people will be grievously disappointed if he does not whip the Government into providing us with a powerful fleet of Zeppelins in about three weeks.
So Colonel Churchill ² has gone back to the Front, and, as Mr. Balfour ³ remarked, “to those opportunities for calm meditation which the Front presents.” It was at the Front, you will remember, that Colonel Churchill’s mind became clear, and he saw that what the country wants is Lord Fisher ⁴ back at the Admiralty; so he came back to tell us so. Probably the ex-First Lord had the most uncomfortable time of his life while Mr Balfour was devoting his attention to him, playing him delicately in his suave, courteous, and best Balfourian manner. Four months ago Mr. Churchill decided that the nation wanted soldiers more than Cabinet Ministers, and he went forth to fight. He was a major then and he is a colonel now. Report speaks of him as an extremely able soldier, very popular with his men. Still, he has hardly such a chance of shining in Flanders as his great ancestor, the first Duke of Marlborough, had, and it appears that, after all, he was half inclined to come back to Westminster. He was, I understand, invited to lead the Liberal “ginger” group, which, like a similar group on the Unionist side, exists for the purpose of infusing energy into the Government. Colonel Chuchill [sic], having given the matter thorough consideration, and talked it over with the Prime Minister, has decided to gird on his sword again.

² The British statesman Winston Leonard Spencer Churchill (1874-1965) served as First Lord of the Admiralty from October 1911 to May 1915.
³ The British statesman Arthur James Balfour (1848-1930) succeeded Winston Churchill as First Lord of the Admiralty in May 1915.
⁴ The Royal Navy officer John Arbuthnot Fisher (1841-1920) served as First Sea Lord from October 1914 to May 1915.