As Francesco says, the smallest example is $D_{10}$. It is a normal subgroup of the Frobenius group of order $20$ and that extension is not split, as can be seen from looking at the $2$-Sylows. There are many more small examples, here is Magma code that will produce them for you:

```
for X in SmallGroups([2..100]) do
auts:=AutomorphismGroup(X);
G:=PermutationGroup(auts);
m:=PermutationRepresentation(auts);
H:=sub<G|[a: a in G | IsInner(a@@m)]>;
if not HasComplement(G,H) then GroupName(X), GroupName(G); end if;
end for;
```

The first few lines of the output:

```
D10 G20
D16 D8:C2^2
Q16 D8:C2^2
C2*D8 C2wrC2^2
C2*Q8 C2^3:S4
C3:S3 H432
C5:C4 C2*G20
D20 C2*G20
D26 C13:C12
C3*D10 C2*G20
C8:C4 (C2^2*C4):D8
C8:C4 C2^2:(D8:C2^2)
C8:C4 C2^2:(D8:C2^2)
H32^15 D8:C2^3
OMC32 D8:C2^2
D32 D16:(C2*C4)
Q32 D16:(C2*C4)
```

Instead of a (mostly) human-readable name, you can also get a unique identifier for the groups by replacing "GroupName" by "IndentifyGroup". That will allow you to play more with these examples, by allowing you to reconstruct them later from their identifiers.