So let's say I have a description of a group, for example:
"extend the group H generated by $\beta$ pairs of non commutative involutions such that all the products of these pairs have a common square of order 2 and that each operator is commutative with every operator of the set except the other one of the pair to which it belongs. The extending operator of order 4 has the same square as the operators of order 4 contained in H."
We created a free group and explicitly made generator relations (see below). Is there an easier way to create a group like that in GAP? I'm thinking more along the lines of following the way they describe generating the group as opposed to having to turn that into generator relations since there's lots of room for potential errors and hence the possibility of ending up with a completely different group doing it that way (not to mention how tedious it is)
There isn't anything approaching what you imagine, basically because it is essentially impossible to get language translated to specific and correct relations. You might be able to write a function for your concrete situation, or get relations by suitable calculations in the free group. As for simplifying typing, you might want to look at the functions
AssignGeneratorVariablesandParseRelatorsand use a free group with specific generators, e.g.FreeGroup("a","b","c");