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. 

<cite authors="Bing, R. H.">_Bing, R. H._, [**A homeomorphism between the 3-sphere and the sum of two solid horned spheres**](http://dx.doi.org/10.2307/1969804), Ann. Math. (2) 56, 354-362 (1952). [ZBL0049.40401](https://zbmath.org/?q=an:0049.40401).</cite>

See also [here](https://lamington.wordpress.com/2017/04/08/bings-wild-involution/) 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.