The immigration debate was at a fever pitch again Wednesday, with groups from different sides of the issue taking aim at House Majority Leader Eric Cantor, a Virginia Republican, who sent a flier to voters in recent days boasting of shutting down a plan to "give illegal aliens amnesty."

The flier, distributed in his district by his re-election campaign, comes as Cantor is under pressure ahead of his June 10 GOP primary — and as the narrow window for action on immigration legislation in the House closes.

At the same, President Obama came under fire for – proponents for more lenient immigration laws say –  backpedaling on a vow to review deportation practices and possibly limit them to dangerous criminals.

Cantor's flier underscores how vexing the issue is for the GOP, seen by critics as the roadblock to comprehensive immigration reform.

Advocates on the left are accusing Cantor of standing in the way of getting overhaul legislation through the House 11 months after the Senate passed a sweeping bipartisan bill.

Cantor's tea party opponent in Virginia's GOP primary, Dave Brat, convened on Wednesday a press conference on the steps of the Virginia Capitol to label Cantor a top cheerleader for "amnesty" in the House, citing Cantor's support for action on certain immigration measures.

A short time later, in a conference room inside the state Capitol, Rep. Luis Gutierrez, D-Ill., a leading proponent of overhauling immigration laws, held his own press conference to accuse Cantor of standing in the way of progress in the House on a measure that would address border security as well as a path to citizenship for the 11.5 million immigrants living in the country illegally.

"Allow America to have a vote," Gutierrez implored. Several immigrants who are facing deportation orders joined him.

Obama, meanwhile, drew the ire of advocates of more lenient immigration policies after his administration’s announcement Tuesday night that it is delaying until late summer the results of a review of the nation's deportation policy.

Advocates had met with Homeland Security Secretary Jeh Johnson in recent weeks and pushed for an end to the high rate of deportations, particularly the detention and expulsion from the country of undocumented immigrants who have no criminal background and are not national security threats.

The administration had indicated that it would review its deportation process and perhaps focus it on dangerous criminals, something it had said several times before but has not fully implemented.

That emboldened House Republicans to argue that they can't trust the current administration to enforce laws and that Obama is undermining prospects for a broader immigration overhaul in Congress.

The White House sees a narrow window this June and July for Congress to act on immigration before the August congressional recess, which is when, traditionally, incumbents begin to seriously focus on midterm elections this November. So Obama directed Johnson not to release the results of the deportation review until that window closes.

Cantor’s flier references Obama and Senate Majority Leader Harry Reid, and says, "Barack Obama and Harry Reid: Pushing amnesty to give illegal aliens a free ride. Conservative Republican Eric Cantor is stopping this liberal plan."

That strong message was a changed tone for Cantor, who has repeatedly voiced support for giving citizenship to certain immigrants brought illegally to the country as children. Last year Cantor joined Gutierrez and other lawmakers on a three-day tour of immigration-related sites in New York aimed at increasing awareness of the issue.

Brat dismissed the new tone from Cantor as politics and flip-flopping.

"Eric Cantor saying he opposes amnesty is like Barack Obama saying he opposes Obamacare," Brat said.

Cantor's campaign spokesman did not immediately return a call seeking comment.

Based on reporting by The Associated Press.

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