We know that given a finite group $G$ and its group 4-cohomology class $w \in H^4[G;U(1)]$, we can obtain a DW topological invariant $Z_{G,w}(M^4)$ as the partition function of the DW theory on a closed 4-manifold $M^4$.
Is it possible that a finite group $G$ may have two different 4-cohomology classes $w$ and $u$, such that they give rise to the same DW topological invariant (ie $Z_{G,w}(M^4) = Z_{G,u}(M^4)$ for any closed $M^4$).
When the group $G$ has an outer automorphism and if $w$ and $u$ are related by the outer automorphism, then $Z_{G,w}(M^4) = Z_{G,u}(M^4)$. Here I would like to ask if there are examples beyond outer automorphism.
In 3-dimensions, there are some dramatic examples beyond outer automorphism that $Z_{G,w}(M^3) = Z_{H,u}(M^3)$ even when $G\neq H$.
