In my question I will use the notation of the lecture notes of my commutative group theory lecture (they are in German).
In the section about group extensions I am at the point of an extension of a nonabelian group $N$ with any group $H$. With the definition of matchings at 4.25 and 4.27 we get that the number of equivilance classes of parameter systems of $N$ with $H$ with some matching $w$ equal the number of equivilance classes of parameter systems of $Z:=Z(N)$ with $H$ with automorphism system $w_Z$.
Now in 4.28(i) we deduce that with $Z=1$ for every matching there is exactly one equivalence class of parameter systems that has this matching. And now I cannot figure out, why in 4.28(ii) we need another assumption for $N$ (namely $Aut(N)$ being a semidirect product of $Inn(N)$ and $Aut(N)/Inn(N)$) to deduce that every extension of $N$ splits. For any parameter system $(a,k)$ the function
$$w_a:H \to Out(N),x \mapsto a_xInn(N)$$
is a homomorphism and thus a matching for $a,k$. Thus, with (i), there is only one equivalence class of parameter systems with $a$ as automorphism system. Thus $(a,k)$ should be in the same equivalence class as $(a,1)$ as for any automorphism system $b$ the parameter system $(b,1)$ exists. Then however $a$ is an homomorphism and the chosen extension is equivalent (and thus isomorphic) to the semidirect product of $N$ and $H$.
This all leads to my question of: At what point did I mess up with my conclusions and why do we need to be able to lift matchings to automorphism systems? For the latter I understood that the lifting guarantees that $Par(w)$ is not empty, but I do not understand why that matters here.
In that context even an example of a non-splitting extension of a group with trivial center with any other group would really help me (I am not very good with central products, but there got to be a good example to find using those). Maybe with that I can find where I am wrong myself.
Thank you in advance for any help, I hope this question is reasonably put.