The most important lobby / activist group about the CSA is NACSA (National Association for Child Support Action):
"NACSA: Here to aid those suffering from the extremes of the CS Agency and to further the promotion of a fair and equitable system of child support".

NACSA run a number of forums concerned with obtaining advice and many other things. (I am a member of NACSA, although childfree). NACSA is undergoing (welcome) reforms - in my words "to be the lobby/support group needed for the 21st Century". Here are some articles about NACSA: The need for a lobby group; NACSA for the 21st Century?

FNF (Families Need Fathers) is a charity which promotes Shared Parenting:
"FNF: Keeping Children and Parents in Contact since 1974".

FNF also frequently makes its views known about the CSA, although that is not its prime focus. They have presented evidence to the Social Security Select Committee about the reform of the CSA. (I helped them prepare & present their case. Part of this analysis was personally rejected by the minister concerned, Baroness Hollis).

The current scheme was heavily influenced by the Treasury. Read, for example:

Child Support In Action
Gwynn Davis, Nick Wikeley, Richard Young with Jacquelin Barron, Julie Bedward
Hart Publishing, 1998
ISBN 1-901362-70-1

Trial and error: a review of UK child support policy
Helen Barnes, Patricia Day, Natalie Cronin
Family Policies Study Centre, 1998
ISBN 1-9011455-08-4

Now the reformed scheme is also being heavily influenced by the Treasury, even though there is far less justification for doing so.

The reformed system is very much a product of politics & administrative concerns, rather than simply satisfying some clear objective about children needing to be supported by both parents.
See the political drivers of the reformed scheme.

The horrors of the current child support system have probably caused the government to over-react.
Here are flaws with treating "administrative ease" with priority.