I am having trouble understanding how to calculate all possible cyclic extensions of a group. I have been following the text 'A Course In Group Theory' by John F. Humphreys, and also referring to some notes Here by Martin Isaacs.
In the notes linked to above, it says that one of the conditions on $ \sigma \in Aut(N)$ is that $ \sigma ^{m}$ is the inner automorphism of $ N $ induced by $ a $ . Am I correct in thinking this means that $\sigma ^{m} $ is the same as conjugating by $a$ ?
If I had a group $G$ with a subgroup $N$ with index $p$ (a prime) , and I wanted to know how many such groups $G$ exist, would calculating all the extensions of $N$ by the cyclic group of order $p$ answer this question?
I am confused about the role of the $a\in N$. To count the number of extensions, would I have to go through each of the $|N|$ possible choices of $a$, and for each $a$ count the number of automorphisms $\sigma \in Aut(N)$ that obey the conditions that: $a^{\sigma}=a$ and $ \sigma ^{m}$ is the inner automorphism of $ N $ induced by $ a $
Could somebody explain the steps I should go through to find all cyclic extensions of a group $N$ by some cyclic group $H$ , and if possible recommend where I can find some worked examples.
Thank you