I know (and understand) how a single group can form a category (of one object, morphisms are the action of the group elements as endomorphisms on it). However here Describing the Wreath product categorically., Qiaochu Yuan states that moreover a group $G$ forms a 2-category, which seems to be quite interesting but lacks of an explanation on the internet unfortunately. Can you please explain which the 2-morphisms are in that case? Also if someone knows, can you write some of the applications of this 2-category?
If you want to understand how the category of groups becomes a 2-category, and in general if you want to understand 2-categories, you ought to be familiar with the canonical example of a (strict) 2-category, namely Cat itself. This the the category of all (small) categories, and has as its objects categories, as its morphisms functors, and as its 2-morphisms natural transformations. Taking a group to be a category with one object, Grp becomes a full subcategory of Cat. Ultimately, I would recommend that you try to spend some more time understanding all these definitions, as once you have that set, the 2-category structure of Grp will become pretty straightforward.