Summary: For every action \(\phi \in \operatorname{Hom} (G, \operatorname{Aut}_k(K))\) of a group \(G\) on a commutative ring \(K\) we introduce two abelian monoids. The monoid \(\mathrm{Cliff}_k (\phi)\) consists of equivalence classes of strongly \(G\)-graded algebras of type \(\phi\) up to \(G\)-graded Clifford system extensions of \(K\)-central algebras. The monoid \(\mathcal{C}_k(\phi)\) consists of equivariance classes of homomorphisms of type \(\phi\) from \(G\) to the Picard groups of \(K\)-central algebras (generalized collective characters). Furthermore, for every such \(\phi\) there is an exact sequence of abelian monoids \[ 0 \to{H^2}(G,K_\phi^\ast) \to \mathrm{Cliff}_k(\phi) \to \mathcal{C}_k(\phi) \to{H^3}(G,K_\phi^\ast). \] This sequence describes the obstruction to realizing a generalized collective character of type \(\phi\), that is it determines if such a character is associated to some strongly \(G\)-graded \(k\)-algebra. The rightmost homomorphism is often surjective, terminating the above sequence. When \(\phi\) is a Galois action, then the well-known restriction-obstruction sequence of Brauer groups is an image of an exact sequence of sub-monoids appearing in the above sequence.