Faithful semidirect product of cyclic p-groups

Definition
This is a general type of group of prime power order obtained as follows. Consider natural numbers $$l < k$$, and an odd prime $$p$$. Now, the multiplicative group of $$\mathbb{Z}/p^k\mathbb{Z}$$ contains a cyclic subgroup of order $$p^l$$: the subgroup generated multiplicatively by $$p^{k-l} + 1$$.

The group we are interested in is the semidirect product of $$\mathbb{Z}/p^k\mathbb{Z}$$ with this cyclic group.

Solvable length
The group is a semidirect product of one cyclic group by another, so it is a metacyclic group. In particular, it is a metabelian group: it has solvable length two.

Nilpotence class
The class of this group depends on $$k,l$$.