In this paper, the notion of an $\mathcal{A}$-group-like unitary system on a Hilbert $C^{\ast }$-module is introduced and some basic properties are studied, where $\mathcal{A}$ is a unital $C^{\ast }$-algebra. Let $\mathcal{U}$ be such a unitary system. We prove that a complete Parseval frame vector for $\mathcal{U}$ can be dilated to a complete wandering vector. Also, it is shown that the set of all the complete Bessel vectors for $\mathcal{U}$ can be parameterized by the set of the adjointable operators in the double commutant of $\mathcal{U}$, and that the frame multiplicity of $\mathcal{U}$ is always finite.