The Naimark--Sz.--Nagy characterization of positive definite functions on groups and Stinespring's decomposition for completely positive maps on a \(C^*\)-algebra are well-known representation theorems. The covariant extension of Stinespring's theorem was given by \textit{V.\,Paulsen} [Mich.\ Math.\ J.\ 29, 131--142 (1982; Zbl 0507.46060)], and a more general version for equivariant completely bounded maps was established by \textit{I.\,Raeburn}, \textit{A.\,M.\thinspace Sinclair} and \textit{D.\,P.\thinspace Williams} [Pac.\ J.\ Math.\ 139, No.\,1, 155--194 (1989; Zbl 0692.46056)]. Furthermore, there are dilations associated with completely multi-positive maps, see [\textit{J.\,Heo}, J.~Oper.\ Theory 41, No.\,1, 3--22 (1999; Zbl 0994.46019)]. The author of the paper under review considers a unified approach to such dilations using the Kolmogorov decomposition for positive definite kernels and investigates the dilations associated with projective \(\sigma\)-representations, which generalize the above dilations. For a strictly \(U\)-covariant and completely positive map from a \(C^*\)-algebra \(B\) into the algebra \(L_A(X)\) of adjointable module maps on a right Hilbert \(C^*\)-module \(X\), the author also constructs a representation of a right Hilbert \(A\)-module.