In "Smooth Homology Spheres and their Fundamental Groups" Kervaire proves that i) every 4-dimensional homology sphere bounds a contractible smooth manifold, and ii) for $d \geq 5$, every $d$-dimensional homology sphere bounds a contractible smooth manifold, after perhaps changing it by connect-sum with an exotic sphere.
Hence, let $\Sigma^d$ be any homology sphere of dimension $d \geq 4$, and modify it if necessary by connected-sum with a homotopy sphere so that it bounds a contractible manifold $\Delta^{d+1}$. Then $$N := \Delta \cup_\Sigma \Delta$$ has an obvious (smooth) involution with fixed set $\Sigma$. Furthermore, this is easily seen to be a homotopy sphere (by Mayer--Vietoris and Seifert--van Kampen). It may not be diffeomorphic to $S^{d+1}$, but by changing $\Delta$ by connect-sum withas it has an exotic sphere $Q$ we change $N$ by connect sum withorientation-reversing involution it will have order $2Q$. So$2$ in those dimensions where the group $\Theta_{d+1}$ of exotic spheres has even order. When (which up to 20$d+1=5,6, 12, 13$ this group is every $d+1$ except 13) we can always re-choosetrivial or $\Delta$$\mathbb{Z}/3$, and so that $N$ ismust in fact be diffeomorphic to $S^{d+1}$ in these cases.
Thus $\Sigma$ need not be a homotopy sphere in general