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By considering the notion of action of a categorical group G on another categorical group H we define the semidirect product H G and classify the set of all split extensions of G by H. Then, in an analogous way to the group case, we develop an obstruction theory that allows the classification of all split extensions of categorical groups inducing a given pair (ϕ, ψ) (called a collective character of G in H) where ϕ : π 0 (G) → π 0 (Eq(H)) is a group homomorphism and ψ : π 1 (G) → π 1 (Eq(H)) is a homomorphism of π 0 (G)-modules. This paper has been financially supported by NATO PST.CLG 975316.

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ -equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.It is well known that the study of groups with operators has many important applications in algebra and topology. The category of groups enriched with an action by automorphisms of a given group provides a suitable setting for the investigation of an extensive list of subjects with recognized mathematical interest. See, for instance, recent results in equivariant stable homotopy theory [6] and articles devoted to equivariant algebraic K-theory [13,29,24]. The origin of the equivariant investigation in homological algebra, particularly in extension theory of groups, goes back to the article of Whitehead [36]. It should be noted that recently a theory of cohomology of groups with operators was developed [8], motivated by E-mail address: hvedri@rmi.acnet.ge (H. Inassaridze).0166-8641/$ -see front matter 

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Abstract. The first non-abelian cohomology of groups introduced by Guin is extended to any dimensions and for a substantially wider class of coefficients called G-partially crossed P-modules. The first and the second non-abelian cohomologies of groups are described in terms of torsors and extensions of groups respectively. Higher non-abelian cohomology pointed sets are described in terms of cotriple right derived functors of the group of derivations with respect to the first contravariant variable. For any short exact coefficient sequence a long exact cohomology sequence is obtained extending the well-known exact cohomology sequences and higher cohomology of groups with coefficients in any G-group is introduced.2000 Mathematics Subject Classification. 18G50, 18G55. 0. Introduction. Our approach to non-abelian cohomology of groups follows Guin's first non-abelian cohomology [5,6] which differs from the classical first nonabelian cohomology pointed set [10] and from the setting of various papers on nonabelian cohomology [4,2,3] extending the classical exact non-abelian cohomology sequence from lower dimensions [10] to higher dimensions.Guin defined his first non-abelian cohomology group when the coefficient group is a crossed G-module and obtained a six term exact cohomology sequence for any short exact sequence of crossed G-modules.A non-abelian cohomology of groups will be defined in any dimension greater than 0, extending Guin's first non-abelian cohomology group and his exact cohomology sequence to a nine term exact cohomology sequence. A substantially wider class of coefficients will be used, consisting of partially crossed modules over a group P on which a group G acts on the left: these will be called G-partially crossed modules over P. We describe the first non-abelian cohomology in terms of torsors and the second non-abelian cohomology in terms of extensions of groups. Moreover a close relation between non-abelian cohomology of groups and nonabelian right derived functors of the group of derivations will be established and for some particular cases of coefficients a long exact cohomology sequence will be obtained.All considered groups will be arbitrary (not necessarily commutative). An action of a group G on a group A means an action on the left of G on A by automorphisms and will be denoted by g a; g 2 G, a 2 A: We assume that G acts on itself by conjugation. The center of a group G will be denoted by ZðGÞ: If the groups G and R act on a group A then the notation gr a means g ð r aÞ, g 2 G; r 2 R, a 2 A:Glasgow Math.