Let $G$ be a finite group with a subgroup $K$. Given $[\beta]\in H^2(K,\mathbb{C}^*)$ is the an obstruction which checks whether or not $[\beta]$ is the restriction of some $[\alpha]\in H^2(G, \mathbb{C}^*)$ (all with trivial action)? Or more generally if it is a restriction of a 2-cocycle from some subgroup in $G$ which contains $K$?
When $G$ is abelian, this question can be reduced to the case where $G$ is a $p$-group because if $G=G_1 \times G_2$ is a product of groups of coprime orders, then $H^2(G,\mathbb{C}^*)=H^2(G_1,\mathbb{C}^*)\times H^2(G_2,\mathbb{C}^*)$ and $K$ must be of the form $K_1\times K_2$ with $K_i \leq G_i$.
Given $G$, a subgroup $K$ and a 2-cocycle on $K$, I can check if it can be extended to some sub group of $G$ using some long algorithm which plays around with the generators of $G$ and $K$ and the values of the cocycle. It mainly asks what roots of unity appear in the cocycle in relation to the order of the elements in the group.
My question is if there is a more direct way of checking that (for an arbitrary group, or at least for a large family of groups).
In case such an obstruction isn't known (or it is too complicated on its own), I would be happy if someone can point me to a database of 2-cocycles on small groups (if there is one).
