Moduli 'space' of stacks?

Such a moduli problem for stacks is expected to be a $2$-stack.

For example, consider the stack of line bundles on $X$, whose objects are parameterized by $H^1(X, \mathbb{G}_m)$. This is a (trivial) example of a $\mathbb{G}_m$-gerbe on $X$; these in turn are parameterized by $H^2(X, \mathbb{G}_m)$. The collection of those forms a $2$-stack.

See the introduction to Lurie's Higher Topos Theory for a nice discussion.

Fix group $G$ (could be a finite group, could be an algebraic group). The collection of all stacks isomorphic to $BG$ is naturally a $2$-stack.

Its $\pi_1$ is $Out(G)$, and its $\pi_2$ is $Z(G)$.

The $k$-invariant in $H^3(Out(G),Z(G))$ which classifies the $2$-stack up to isomorphism is the obstruction to the existence of a short exact sequence of groups $$ 0 \to Z(G) \to E \to Out(G) \to 0 $$

It's not very easy to come up with an example of a group $G$ where the obstruction is non-zero (by which I mean it's not very easy to compute the obstruction). I seem to remember that the obstruction is non-zero for the dihedral group with $16$ elements.