A new battle has flared inside the Republican Party in recent days as supporters of more-liberal immigration laws wage a behind-the-scenes campaign to discredit the influential advocacy groups that have long powered the GOP’s hard-line stance on the issue. …

Now, Republicans pushing the party to rethink its approach to the issue are accusing those groups — Numbers USA, the Center for Immigration Studies (CIS), and the Federation for American Immigration Reform (FAIR) — of masquerading as conservative. Critics say the groups and some of their supporters are pressing an un­or­tho­dox agenda of strict population control that also has included backing for abortion, sterilization and other policies at odds with conservative ideology.

“If these groups can be unmasked, then the bulk of the opposition to immigration reform on the conservative side will wither away,” said Alfonso Aguilar, executive director of the Latino Partnership for Conservative Principles and a leading organizer of the effort.

Officials from the groups say they are the victims of a smear campaign that unfairly characterizes their mission. …