Summary: We consider positive definite kernels which are invariant under a multiplier and an action of a semigroup with involution, and construct the associated projective isometric representation on a Hilbert \(C^*\)-module. We introduce the notion of \(C^*\)-valued Hilbert-Schmidt kernels associated with two sequences and construct the corresponding reproducing Hilbert \(C^*\)-module. We also discuss projective invariance of Hilbert-Schmidt kernels. We prove that the range of a convolution type operator associated with a Hilbert-Schmidt kernel coincides with the reproducing Hilbert \(C^*\)-module associated with its convolution kernel. We show that the integral operator associated with a Hilbert-Schmidt kernel is Hilbert-Schmidt. Finally, we discuss a relation between an integral type operator and a convolution type operator.