If $G$ is a discrete group, recall that a (naive) $G$-spectrum consists of based $G$-spaces $E_n$ together with based $G$-maps $\Sigma E_n \to E_{n+1}$, where we give the suspension coordinate the trivial action. A standard example is to take an abelian group $A$ with a $G$-action, i.e., a $G$-module structure. Then the Eilenberg-Mac Lane spectrum $hA$ has the structure of a $G$-spectrum. Notice that taking path components of $hA$ recovers $A$ together with its module structure.

Suppose we are given a extension of groups $$ 0 \to A \to E \to G \to 1 $$ where $A$ is any abelian group. Then we have an associated $G$-module structure (the action is given by lifting elements of $G$ to $E$ and then conjugating).

So we have an operation $$ \lbrace \text{extensions of } G \text{ by } A \rbrace \qquad \mapsto \qquad \lbrace G\text{-module structures on } A \rbrace . $$ The problem is to now fill in the diagram $$ \lbrace \quad\qquad\text{ ?}\qquad \quad \rbrace \quad \mapsto \qquad \lbrace G\text{-action on a spectrum } E \rbrace $$ $$ \downarrow\qquad\qquad \qquad \qquad\qquad \qquad \downarrow $$ $$ \lbrace \text{extensions of } G \text{ by } A\rbrace \qquad \mapsto \qquad \lbrace G\text{-module structures on } A\rbrace . $$ What is the appropriate notion of extension of $G$ by a spectrum?

After all, a spectrum is something akin to "an abelian group up to homotopy."

`a fiber bundle over $BG$ with fibers $A$'' (a bundle of spectra). But it is a variation of the same`

short and lazy'' answer. $\endgroup$