The best replacement for invite is invitation.

The best alternatives to invite are rated as following:
★★★★★  invitation
★★  invites
★  cordially invites
★  who to invite
  write
  reinvite
  envite

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Speaking about invite, one probably shouldn't forget about gmail.

Other search terms related to invite and their ratings are:
★★★★★ gmail
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Google Trends

Scale is based on the average worldwide traffic of invite in all years. Click on graph for more.
invite
invitation
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