Let $M^3$ denote the Poincare homology sphere. I am wondering what the possible orders of (smooth) automorphisms of $M$ are (I'm not sure if allowing arbitrary homeomorphisms changes things?). By presenting the space as $-1$-surgery on the trefoil, there is a automorphism of order 3 given by the 3-fold symmetry of the trefoil. Similarly, the framed link description of $M$ obtained by presenting $M$ as the 5-fold cyclic branched cover of $S^3$ branched over the trefoil has a 5-fold symmetry. Both of these symmetries can also be seen from the dodecahedron/icosahedron definition of $M$.

I believe that the $5k$-fold cyclic branched covers of $S^3$ branched over the trefoil are all $M$ for any $k \geq 1$ and therefore, by drawing the resulting framed link representations, we obtain automorphisms of order $5k$ for any $k$. I am not sure if any of these automorphisms are isotopic to the identity.

What are the possible orders of automorphisms of $M$? Which possible orders have an automorphism of that order that is not isotopic to the identity?