I don't know about any group activities either I'm afraid, but how about a bus ride on the 344 to Battersea and then to the Children's zoo in Battersea Park - our friend's 4yr old enjoyed both. The zoo did do an Easer egg hunt earlier so maybe they put on other events during the summer school hols too.
The cafe in the Park is good for a coffee/juice + cakes etc too, oh and lake + boats for hire will be open then too.
The thing is for them to meet and socialise with other children - in ENGLISH. They know English, but are used to speaking Hebrew and they find it to easy to talk to my daughter and me in Hebrew because they know we speak and understand it, even if we resolutely speak English to them. However with other kids they would have no choice, hence the search for a GROUP or, otherwise, a bunch of playmates!
And...brain going very slowly today - I think the new Coin Street community centre (nr Waterloo) do workshops and similar for young children - not sure of their web address - but I'm sure if you just google Coin Street you'll find them.
Try Arty Party in Clapham - brilliant and you can do a minimum of 2 days but book early - or Bed Heads Football in Battersea. Both accept younger children.
Look also at websites of Unicorn, Young Vic, Globe, childrens concerts/workshops at the RFH, War Museum, Florence Nightingale M, Garden Museum (they do a lot, as well as Vauxhall and Surrey Quays City Farms and Roots and Shoots and of course the Tate.
And James is right. There is always lots going on and his listing is invaluable.