Suppose you have a finite group $G<SO(5)$ such that $S^4/G$ is homeomorphic to $S^4$ and such that $S^4/G$ is a PL manifold with respect to a PL structure induced by a standard structure on $S^4$. Is there a general argument by which $S^4/G$ is a standard PL $4$-sphere?

Example: Take $G$ to be the exotic image of the alternating group $A_5$ in the symmetric group $S_6<SO(5)$. In this case one can check by hand that $S^4/G$ is a standard PL $4$-sphere, but in the way I do it it is very cumbersome.