A good place to look for finite toy examples is in the theory of group extensions with non-abelian kernel.

A homomorphism $G \to Out(H)$ is the same as a functor $G \to Ho(Top)$ sending $*$ to $BH$, where $G$ is viewed as a category with one object $*$ and $G$ as morphisms. By the Dwyer-Kan obstruction theory (noting that the diagram is centric) the obstructions to lifting this to a diagram in Top lies in $H^3(G;Z(H))$.

A closer inspection shows that this obstruction class is the same as the obstruction in $H^3(G;Z(H))$ to constructing an extension

$$1 \to H \to ? \to G \to 1$$

It is possible to calculate this obstruction group in a number of cases:

Eilenberg-MacLane show in the 1947 Annals paper "Cohomology Theory in Abstract Groups II" that given a group $G$ and an abelian group $C$ with an action of $C$, and a class $v$ in $H^3(G;C)$ then there exist a group $H$ with center $C$ and a morphism $G \to Out(H)$ realizing the class $v$. (A special case is proven in MacLane's book Homology, and there is an exposition on some of this in Brown's book on group cohomology.) Note that in this construction $H$ may be infinite even if $G$ and $C$ are not.

It is also possible to give completely finite examples: Given a group H there is a universal obstruction in $H^3(Out(H);Z(H))$ associated to this group, determining whether the extension

$$1 \to H \to ? \to Out(H) \to 1$$

exists. Other obstruction classes will be a pull-back of this one (in particular it has the best chance of being non-zero).

The topological interpretation of this obstruction class is that it is the unique k-invariant of the space $BAut(BH)$, where Aut means the space of self-homotopy equivalences. (Recall that group extensions correspond to fibrations, and fibrations with fiber $F$ are classified by maps into $BAut(F)$, and this is a good way to understand group extensions with non-abelian kernel.)

Now, it is possible to calculate this class explicitly for small groups $H$. It of course vanishes when H is abelian. It also turns out to vanish for $H = D_8$ or $Q_8$, but is actually non-zero for $D_{16}$ and $Q_{16}$.

[As an aside I can say that the H^3 class always vanishes for compact connected Lie groups, which is a theorem of de Siebenthal from the 1950's. Kasper Andersen and I proved that this generalizes to p-compact groups (in a G&T paper from 2008), which was why Kasper and I examined the case where $H$ finite.]

[As a further aside I can say that there are a lot of similar obstruction questions over relatively small categories such as the orbit category and corresponding centric diagrams, which occur in e.g., in the theory of p-compact groups and p-local finite groups, when addressing certain uniqueness questions -- here lucily the obstructions are usually zero, although this is usually not so easy to prove, and in particular don't seem to follow from "formal" arguments.]

A good place to look for finite toy examples is in the theory of group extensions with non-abelian kernel.

A homomorphism $G \to Out(H)$ is the same as a functor $G \to Ho(Top)$ sending $*$ `to $BH$, where $G$ is viewed as a category with one object $*$ and $G$ as morphisms. By the Dwyer-Kan obstruction theory (noting that the diagram is centric) the obstructions to lifting this to a diagram in Top lies in $H^3(G;Z(H))$.

A closer inspection shows that this obstruction class is the same as the obstruction in $H^3(G;Z(H))$ to constructing an extension

$$1 \to H \to ? \to G \to 1$$

It is possible to calculate this obstruction group in a number of cases:

Eilenberg-MacLane show in the 1947 Annals paper "Cohomology Theory in Abstract Groups II" that given a group $G$ and an abelian group $C$ with an action of $C$, and a class $v$ in $H^3(G;C)$ then there exist a group $H$ with center $C$ and a morphism $G \to Out(H)$ realizing the class $v$. (A special case is proven in MacLane's book Homology, and there is an exposition on some of this in Brown's book on group cohomology.) Note that in this construction $H$ may be infinite even if $G$ and $C$ are not.

It is also possible to give completely finite examples: Given a group H there is a universal obstruction in $H^3(Out(H);Z(H))$ associated to this group, determining whether the extension

$$1 \to H \to ? \to Out(H) \to 1$$

exists. Other obstruction classes will be a pull-back of this one (in particular it has the best chance of being non-zero).

The topological interpretation of this obstruction class is that it is the unique k-invariant of the space $BAut(BH)$, where Aut means the space of self-homotopy equivalences. (Recall that group extensions correspond to fibrations, and fibrations with fiber $F$ are classified by maps into $BAut(F)$, and this is a good way to understand group extensions with non-abelian kernel.)

Now, it is possible to calculate this class explicitly for small groups $H$. It of course vanishes when H is abelian. It also turns out to vanish for $H = D_8$ or $Q_8$, but is actually non-zero for $D_{16}$ and $Q_{16}$.

[As an aside I can say that the H^3 class always vanishes for compact connected Lie groups, which is a theorem of de Siebenthal from the 1950's. Kasper Andersen and I proved that this generalizes to p-compact groups (in a G&T paper from 2008), which was why Kasper and I examined the case where $H$ finite.]

[As a further aside I can say that there are a lot of similar obstruction questions over relatively small categories such as the orbit category and corresponding centric diagrams, which occur in e.g., in the theory of p-compact groups and p-local finite groups, when addressing certain uniqueness questions -- here lucily the obstructions are usually zero, although this is usually not so easy to prove, and in particular don't seem to follow from "formal" arguments.]