Suppose Zp acts freely on Sn×Dk with fixed set Fn a stably parallelisable [m/2]-connected manifold with boundary having the Z(p)-homology of a sphere Sm (0<m<n). Assuming some restrictions on the integers k,m,n,p, the author proves that there exists a Zp-invariant homotopy n-sphere Σ⊂Sn×Dk (with nonzero degree modp) such that ΣG⊂Fn is a homotopy m-sphere (with nonzero degree  modp ).