Let $G$ be a finite group. A *categorical Schur detector* for $G$ is a set $\mathcal{S}$ of proper subgroups $S \subsetneq G$ such that the total restriction map
$$ \mathrm{rest}_{\mathcal{S}} : \mathrm{H}^3(G; \mathrm{U}(1)) \to \prod_{S \in \mathcal{S}} \mathrm{H}^3(S; \mathrm{U}(1)) $$
is an injection. The name is a riff on the suggestion of Epa and Ganter to call $\mathrm{H}_3(G;\mathbb{Z})$ the *categorical Schur multiplier* of $G$ (Platonic and alternating 2-groups, *Higher Structures* 1(1):122–146, 2017).

I know of two classes of groups which do not admit categorical Schur detectors: the cyclic and the binary dihedral (aka dicyclic) groups of prime power order. (Of course, a binary dihedral group of prime power order is necessarily of order $2^n$.) This follows from the fact that for any finite subgroup $G \subset \mathrm{SU}(2)$, there is an isomorphism $\mathrm{H}^3(G;\mathrm{U}(1)) \cong \mathbb{Z}/\lvert G\rvert$.

Question:Are there any other finite groups which do not admit categorical Schur detectors?

Of course, one could ask the same question about cohomology of other degrees, or with other coefficients. $\mathrm{H}^3(-;\mathrm{U}(1))$ is of particular interest in "moonshine", because it is home to global symmetry anomalies of 2D QFTs.

2more comments