 Order-sixteen Gassmann pair

Embed $C_4\times C_4$ and $Q_8\times C_2$ in $S_{16}$ by their regular representations. Both have one element of order one, three of order two, and twelve of order four. An order-$r$ element in a regular action has cycle type $r^{16/r}$. Their intersections with every conjugacy class of $S_{16}$ consequently have the same size, so they are Gassmann equivalent. The first group is abelian and the second is not, so their underlying groups are not isomorphic. Finite-cover constructions from the Sunada theorem therefore give isospectral manifolds with different fundamental groups.