Given a positive integer \(q\), in this paper the authors investigate some more properties of \(q\)-capability of groups. For instance, a relationship between \(q\)-capability and varietal capability is determined. The notion of \(q\)-capability was introduced by G. J. Ellis [J. Lond. Math. Soc., II. Ser. 51, No. 2, 243–258 (1995; Zbl 0829.20073)], while that of varietal capability appeared in [M. R. R. Moghaddam and S. Kayvanfar, Algebra Colloq. 4, No. 1, 1–11 (1997; Zbl 0877.20016)]: for a variety \(\nu\) of groups defined by the set of laws \(V\), a group G is \(\nu\)-capable if there exists a group \(E\) such that \(G \cong E/V^*(E)\), where \(V^*(E)\) denotes the marginal subgroup of \(E\). Then one has that \(q\)-capability, for \(q\geq 0\) and \(q \neq 1\), implies \(\nu_q\)-capability, where \(\nu_q\) denotes the variety defined by the set of laws \(V_q = \{x_1^q, [x, y]\}\) (Lemma 3.1). The authors also introduce the notion of \(q\)-epicenter for a group, extending to all \(q \geq 0\) the concept of epicenter introduced in [F. R. Beyl et al., J. Algebra 61, 161–177 (1979; Zbl 0428.20028)], and use it to obtain some criteria for the \(q\)-capability of groups. Finally, as an application, they characterize all \(q\)-capable extra-special \(p\)-groups when \(q\) is a power of \(p\). The structure of the \(q\)-tensor and \(q\)-exterior squares of a group, particular cases of the more general constructions of non-abelian tensor and exterior products modulo \(q\) introduced by D. Conduché and C. Rodríguez-Fernández [J. Pure Appl. Algebra 78, No. 2, 139–160 (1992; Zbl 0795.20035)], plays a fundamental role in the present context.