Given $\Bbb ZQ$-module $K$, how to construct $\theta:Q\to\text{Aut}(K)$?

Question

I have a problem understanding the meaning of the "converse" part of the below proposition. Let $K,~Q$ be groups, and $K$ is abelian. If $K$ is a module over $\Bbb ZQ$, then it can induce a homomorphism $\theta:Q\to\text{Aut}(K)$. However, what does $\theta$ look like? The term "$xa$" in the book made me confused -- how can $x\in Q$ and $a\in K$ be multiplicated?

Answer 1

how can $x\in Q$ and $a\in K$ be multiplicated?

The answer lies in

If $K$ is a module over $\Bbb ZQ$

By using the $\Bbb ZQ$-module structure of $K$, and noting that $Q$ is contained as a subset of $\Bbb ZQ$, namely all different possible $1\cdot x$ for $x\in Q$.