Extending a 2-cocycle on $K \subseteq G$ to a 2-cochain on $G$

Question

Let $K \subseteq G \to G/K$ be an exact sequence of finite abelian groups. Moreover, let $f:G \times G \to \mathbb{C}^\times$ be a bicharacter with $f(g,k)f(k,g)=1$ for all $k\in K, g \in G$ and $\sigma \in Z^2(K,\mathbb{C}^\times)$ a 2-cocycle (trivial action) on $K$, s.t. $f|_{K\times K}=\sigma\cdot(\sigma\circ\tau)^{-1}$, where $\tau$ swaps the arguments. Is there a 2-cochain $\tilde{\sigma}\in C^2(G,\mathbb{C}^\times)$, s.t. $\tilde{\sigma}|_{K \times K} = \sigma$ and the above equation holds on $K \times G$ and $G\times K$?