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I have looked at the other two (or three) threads on this, and the pattern is clear: Treat "push" as a free bull rush on a hit or deny the push ability.
As most players with internet access know there are multiple SRDs/PRDs/Spellbook programs/Charcter builders out there, and there is no consistency on the spell between them--d20pfsdrd.com (PFRPG SRD) has the spell without "push"; archivesofnethys.com (Archieves of Nethys SRD) and thegm.org/alpha/ (Perram's Spellbook PDF creator) both have it with "push."
I think that without an official ruling--though "push is not part of the spell in the 5th printing of PF Core (though it is not identified as an alchemist spell either)--it's going to have to remain on a DM-to-DM basis.