Realization-obstruction exact sequences for Clifford system extensions

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