The smallest example when $\eta: \Pi\to \text{Out}(G)$ does not give rise to an extension
of $\Pi$ by $G$ is when $G=D_{16}$,
the dihedral group has order 16. (Apologies to those who write it as $D_8$.)
Pick $c,h\in D_{16}$
a rotation of order 8 and a reflection.
The elements of $\text{Out}(D_{16})\cong C_2\times C_2$
correspond to four types of automorphisms
$$
\begin{array}{cccccccc}
I &:&& c\mapsto c^{\pm 1} && h\mapsto h\cdot c^{\text{even}} && \text{(inner)}\cr
II &:&& c\mapsto c^{\pm 3} && h\mapsto h\cdot c^{\text{even}} \cr
III &:&& c\mapsto c^{\pm 1} && h\mapsto h\cdot c^{\text{odd}} \cr
IV &:&& c\mapsto c^{\pm 3} && h\mapsto h\cdot c^{\text{odd}}, \cr
\end{array}
$$
and there is no extension of $\Pi=C_2$ by $G=D_{16}$
with $\Pi$ acting on $G$ as a type $IV$ automorphism.
One way to see this is just to go through the 51 groups of order 32;
only five of them contain $D_{16}$, number 18,19,39,42,43 in GAP
or Magma. The first two $D_{32}, SD_{32}$ give a type III action, next two
$C_2\times D_{16}, (C_2\times C_8):C_2$ a type I action, and the last one $D_8:C_2^2$ a type
II action.
Another way is to use the fact that dihedral, semi-dihedral and generalized quaternion
groups are the only non-abelian 2-groups whose commutator subgroup has index 4 (and index 1 or 2 is impossible). The
commutator subgroup $G'$ of $G=D_{16}$ is generated by $c^2$, and an extension $B$ of
type $III$ or $IV$ acts non-trivially on $G/G'=\{1,c,h,ch\}$, so $B'<B$ has again index 4. So
it must be $D_{32}$ or $SD_{32}$, since $Q_{32}$ has no dihedral subgroups. But both of these
are visibly not type $IV$ extensions, because the non-trivial coset of $G$ in $B$ is represented by an element whose square is $c$, and it conjugates $c$ to itself.
I suppose in this example
$$
H^3(\Pi,Z(G))=H^3(C_2,C_2)=H^1(C_2,C_2)=\text{Hom}(C_2,C_2)=C_2,
$$
which pins down the obstruction class uniquely.
