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We present some applications of strict graded categorical groups to the construction of the obstruction of an equivariant kernel and to the classification of equivariant group extensions which are central ones. The composition of a graded categorical group and an equivariant group homomorphism is also determined.

The group extension problem has an important significance in the development of modern algebra. Some notions of this problem such as crossed product, factor set, and obstruction (see [

The theory of graded categorical groups studied by Cegarra et al. [

Strict graded categorical groups, with their simple structures compared to the general case, are more likely to give a lot of interesting applications. In [

We recall briefly some basic notions about graded categorical groups in [

We regard the group

a stable

natural isomorphisms of grade 1

A

If

The authors of [

the set

the set

the third invariant is an equivariant cohomology class

Based on these data, they constructed a

Objects of

The composition of two morphisms

The graded tensor product

The unit constrains are strict in the sense that

The stable

The unit graded functor

It is well known that each crossed module of groups can be seen as a strict categorical group (see [

Firstly, if

A graded categorical group

Equivalently, a graded categorical group

In this paper we denote by + for the operation of the group

The

Firstly, observe that if

For each

The graded tensor product is

The notion of

Let

A

From the definition of

The strict graded categorical group

Objects of

The tensor product on objects is given by the multiplication in the group

The associativity and unit constraints of the tensor product are strict.

The graded functor

The notion of

Theorem 4.1 in [

The following theorem describes the invariants of the graded categorical group

Let

(i) It is obvious.

(ii) According to Section 3 in [

Since

The compatibility of

Next, we show that

Since

Due to the relations (

Let us note that

Finally, we prove that

Denote by

Let

Let

Conversely, suppose that

Now, two extensions in

Conversely, if

Finally, each central extension

It is well known that for a given extension

Let

The strict graded categorical group

The tensor products on objects and on morphisms in

For each morphism

The unit object of

Suppose that the graded categorical group

Thus,

We define a pair

The graded monoidal functor

Suppose that

Finally, we prove that

If

The graded categorical group