Is there a candidate for a post-quantum one-way group action?
Yes, there is an old proposal for this due to Couveignes, which was independently rediscovered by Rostovtsev and Stolbunov.

In both cases, the set of elliptic curves with some common endomorphism ring $\mathcal O$ is acted on by the ideal class group of $\mathcal O$. The secret key is essentially a description of an isogeny via its kernel ideal, and the action of a group element $[\mathfrak a]$ takes a curve $E$ to the codomain of said isogeny:
$$
    ([\mathfrak a], E) \longmapsto E/\mathfrak a = E/\bigcap_{\alpha\in\mathfrak a}\ker\alpha
    \text.
$$
There is a nice graph-walking interpretation of this action, which is described (for instance) in Section 14.1 of Luca De Feo's lecture notes. (They also contain more of the background necessary to understand this construction!)

While it is possible to invert the group action by solving an instance of the hidden-shift problem, giving rise to a subexponential quantum attack, the system remains unbroken for reasonably big parameter sizes.
A much bigger problem is that these schemes are painfully slow in practice: Even after considerable optimization effort, one computation of the group action still takes minutes.

The performance issue has been tackled by a recent proposal called CSIDH by switching to supersingular elliptic curves, which drastically improves efficiency while keeping essentially the same underlying structure. It is still slow relative to comparable pre-quantum schemes, as well as incomparable post-quantum schemes, but may have a place in a post-quantum world due to its unique features.