As the question shows, I want to know this term's meaning. It is from a book Groups of prime power order, Volume 2, page237. The picture is:
And what is the meaning of 'cover' which is the 5th line in the first picture as it shows in the picture?
The group of Lemma65.1 is metacyclic minimal non-abelian p-groups. As it shows as follows:
I believe the following sentence, in your picture, gives a description of a splitting metacyclic group. First of all, a group $G$ is metacyclic if it is an extension of two cyclic groups. That is, there are cyclic groups $A$ and $B$ such that $$ 1 \longrightarrow A \longrightarrow G \longrightarrow B\longrightarrow 1$$ is a short exact sequence of groups. Such a sequence has a notion of split, which is where you get the definition of a split metacyclic group: a group which is a split extension of cyclic groups.
There are other ways to think about split extensions, and the picture given shows that $G/G_0$ is a split metacyclic group: $G=AB$, $A\cap B=G_0$, and $A\triangleleft G$. Thus, $$ 1 \longrightarrow A/G_0 \longrightarrow G/G_0 \longrightarrow B/G_0\longrightarrow 1$$ is a split exact sequence of groups.
Suppose $K\triangleleft H \leq G$ and $L\leq G$. A group $L$ covers a quotient $H/K$ if $HL=KL$.