SIKE as Diffie-Hellman exchange SIKE involves a Diffie-Hellman like construction using isogenies between elliptic curves. Is it possible to express it as an instance of Diffie-Hellman? I.e. what is the underlying group if so, and further, why wouldn't Shor's algorithm apply and render it vulnerable?
I am one of the inventors of SIKE.

There is no actual Diffie-Hellman group in SIKE. If there were, then SIKE would be vulnerable to Shor's algorithm, as you said, largely defeating the purpose of a post-quantum cryptosystem.

Among other reasons, isogenies typically have a domain different from their codomain, so an isogeny cannot be iterated. You can compose two different isogenies (provided that their respective domains and codomains are compatible), but usually not an isogeny with itself.