Twenty other groups, including Consumer Action, Public Citizen and the U.S. Public Interest Research Group, joined in the latest complaint.
Google issued a statement emphasizing its commitment to protecting children while they are online one of the reasons the company says it prohibits targeted advertising at children under 13.
"We take these issues very seriously and continue to work hard to remove any content that is inappropriately aimed at children from our platform," Google said.
More than 2 billion devices worldwide are powered by Google software, with a significant number of those being used by minors. The complaint focuses on alleged misconduct under U.S. laws and regulations.
The attempt to pressure the FTC to open an investigation comes amid an intensifying backlash against Google, Facebook and other companies that make most of their money by using their free services to track people's interests and whereabouts and then mining that information to sell ads targeted at them.
The angst has raised the specter of Congress drawing up tougher regulations to curb the tech industry's power and restrict its ability to compile digital dossiers about the people who have become increasingly dependent on its services.
Rep. David Cicilline, a Democrat from Rhode Island who has been critical of Google, issued a statement supporting the groups seeking an FTC investigation as did Sen. Tom Udall, a Democrat from New Mexico.
"It is past time for the Federal Trade Commission to crack down to protect children's privacy," Udall said in a statement.
Although the FTC doesn't typically comment on whether it will investigate issues raised in complaints, it has punished both Google and Apple for what it deemed to be child exploitation in the past.
In 2014, it reached a settlement requiring Google to refund $19 million for allowing apps distributed through its store to charge children for purchases made without parents' consent.
That came after a similar agreement required Apple to refund $32.5 million for in-app purchases made on iPhones, iPads and other devices without parents' permission.
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