The best replacement for admit is acknowledge.

The best alternatives to admit are rated as following:
★★★★★  acknowledge
★★★★★  admit
★★★★  admit to
★★★★  deny
★★★★  accept
★★  concede
★★  no contest
★★  confess
  permit

Your alternative for admit is not listed? Suggest!
Speaking about admit, one probably shouldn't forget about gerund.

Other search terms related to admit and their ratings are:
★★★★★ gerund
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Google Trends

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admit
acknowledge
admit