Projectivity and injectivity of $\mathsf{G}$-Hilbert $\Im$-modules

‎Let $\mathsf{G}$ be a discrete group acting on $C^*$-algebra $\Im$‎. ‎In this paper‎, ‎we investigate projectivity and injectivity of $G$-Hilbert $\Im$-modules and study the equivalent conditions characterizing $\mathsf{G}$-$C^*$-subalgebras of the algebra of compact operators on $\mathsf{G}$-Hilbert spaces in terms of general properties of $\mathsf{G}$-Hilbert $\Im$-modules‎. ‎In particular‎, ‎we show that $\mathsf{G}$-Hilbert $\Im$-(bi)modules on $\mathsf{G}$-$C^*$-algebra of compact operators are both projective and injective‎.