Vtyjkyuk

We'll take the following definition of a 2-group:

A 2-group $mathsfG$ is a category internal to $mathsfGrp$

Namely, it is a group $mathsfG_0$ of objects, a group $mathsfG_1$ of morphisms, together with maps:

$s,t:mathsfG_1rightarrow mathsfG_0$ (source and target map)

$id:mathsfG_0rightarrow mathsfG_1$ (the identity map)

$circ: mathsfG_1times_(s,t)mathsfG_1rightarrow mathsfG_1$ (composition map between composable morphisms)

such that the usual diagrams defining a category commute.

One way I see to view a 2-group as a 2-category is to say that a 2-group defined as previously is a monoidal category with the group composition as the tensor product. The delooping category $BmathsfG$ is thus the manner to view a 2-group as a 2-category, am I right?

Have a look at what a 2-groupoid should be, then look at a 2-groupoid having just one object. Call the set of objects, $X_0$, take $X_1$, the set of 1-arrows (check this should be a groupoid), then $X_2$ as set of 2-arrows. Write down the axioms you expect, then specialise to look at the structure at a single object. You get a 2-group. (Now reverse engineer the 2-groupoid from the 2-group.)

What you say about a 2-group as a (strict) monoidal category is also correct.

Maybe the definition you gave is not the most suitable to catch the 2-categorical nature of 2-groups: you are probably interested in this pdf.

Hope it helps!