  We extend the notion of face rings of simplicial complexes and simplicial
posets to the case of finite-length (possibly infinite) simplicial posets with
a group action. The action on the complex induces an action on the face ring,
and we prove that the ring of invariants is isomorphic to the face ring of the
quotient simplicial poset under a mild condition on the group action. We also
identify a class of actions on simplicial complexes that preserve the
homotopical Cohen-Macaulay property under quotients. When the acted-upon poset
is the independence complex of a semimatroid, the $h$-polynomial of the ring of
invariants can be read off the Tutte polynomial of the associated group action.
Moreover, in this case an additional condition on the action ensures that the
quotient poset is Cohen-Macaulay in characteristic 0 and every characteristic
that does not divide an explicitly computable number. This implies the same
property for the associated Stanley-Reisner rings. In particular, this holds
for independence posets and rings associated to toric, elliptic and, more
generally, $(p,q)$-arrangements. As a byproduct, we prove that posets of
connected components (also known as posets of {layers}) of such arrangements
are Cohen-Macaulay with the same condition on the characteristic.
