There are involutions $\sigma$ of the 3-sphere, whose fixed-point sets are wild 2-spheres: The fixed-point set cannot be a subcomplex of any triangulation, hence, $\sigma$ cannot be PL in any triangulation.

Bing, R. H., A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Ann. Math. (2) 56, 354-362 (1952). ZBL0049.40401.

See also here for Calegari's take on Bing's proof.

There are involutions $\sigma$ of the 3-sphere, whose fixed-point sets are wild 2-spheres: The fixed-point set cannot be a subcomplex of any triangulation, hence, $\sigma$ cannot be PL in any triangulation.

Bing, R. H., A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Ann. Math. (2) 56, 354-362 (1952). ZBL0049.40401.

See also here for Calegari's take on Bing's proof.

Edit. There is even (unique in some sense) free involution of the 4-sphere which cannot preserve a triangulation (this is due to Ruberman). Thus, a bad fixed point set is not the only obstruction.