Suppose $S^3$ is PL sphere on which a finite group $G$ acts by PL homeomorphisms. Is it always possible to find a compatible smooth structure such that $G$ acts by diffeomorphisms?


I am not quite sure how smoothing works for trivial $G$, but maybe this can be generalized to the situation described above?



Equivariant smoothing of PL structures on $S^3$
I think the answer to your question is "always". You may want to look at the following paper: 


Kwasik, Sławomir; Lee, Kyung Bai. Locally linear actions on $3$-manifolds. Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 2, 253--260. MR0948910. 


In particular, Corollary 2.2: *A topological action of a finite group G on a closed 3-manifold M is smoothable if and only if it is simplicial (in some triangulation of M).*


