In this work, we study a sub-collection of unital completely positive maps from a unital \(C^*\) -algebra \(\mathcal {A}\) to \(\mathcal {B}(\mathcal {H}),\) the algebra of bounded linear operators on a Hilbert space \(\mathcal {H}\) in the setting of \(C^*\) -convexity. Let \(\tau \) be an action of a group G on the \(C^*\) -algebra \(\mathcal {A}\) through \(C^*\) -automorphisms. We focus our attention to the set of all unital completely positive maps from \(\mathcal {A}\) to \(\mathcal {B}(\mathcal {H}),\) which remain invariant under \(\tau .\) We denote this collection by the notation \({\text {UCP}}^{G_\tau } \big (\mathcal {A}, \mathcal {B} (\mathcal {H} ) \big ).\) This collection forms a \(C^*\) -convex set. We characterize the set of \(C^*\) -extreme points of \({\text {UCP}}^{G_\tau } \big (\mathcal {A}, \mathcal {B} (\mathcal {H} ) \big ).\) Further, we conclude the article by proving the Krein–Milman type theorem in the setting of \(C^*\) -convexity for the set \({\text {UCP}}^{G_\tau } \big (\mathcal {A}, \mathcal {B} (\mathcal {H} ) \big ).\)
