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).

The answer is "always", but it is not proven in the paper by Kwasik and Lee. They only show the existence of an equivariant smooth structure which is not necessarily compatible with the PL structure. The statement holds for manifolds up to dimension four. I tried to write a readable proof and put it on the arXiv.