  A group is said to be capable if it is the central factor of some group. In
this paper, among other results we have characterized capable groups of order
$p^2q$, for any distinct primes $p, q$, which extends Theorem 1.2 of S. Rashid,
N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, {\em On the non abelian tensor
square and capability of groups of order $p^2q$}, Arch. Math., \textbf{97}
(2011), 299--306. We have also computed the number of distinct element
centralizers of a group (finite or infinite) with central factor of order
$p^3$, which extends Proposition 2.10 of S. M. Jafarian Amiri, H. Madadi and H.
Rostami, {\em On $F$-groups with the central factor of order $p^4$}, Math.
Slovaca, \textbf{67} (5) (2017), 1147--1154.
