Classification of finite p-groups with self-centralizing cyclic normal subgroup

This article gives a classification statement for certain kinds of groups of prime power order, subject to additional constraints.
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Statement

A finite $p$-group (a group of prime power order) $P$, with a self-centralizing cyclic normal subgroup $H$, could be either of the following:

• If $p$ is an odd prime, it is a faithful semidirect product of cyclic p-groups: it is the semidirect product of $H$ with a cyclic subgroup of the automorphism group of $H$.
• If $p=2$, there are a number of possibilities: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
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Related facts

• Classification of finite p-groups with cyclic maximal subgroup
• Classification of metacyclic p-groups