You cannot add a new group that I know of. You can only change the attributes of the feature type - cross-hatch.
The idea is that all standards recognize the line weight of cross hatch to be the same as dimension lines and text. So from day one, cross hatch was tied in with lettering when it came to assigning pens.
Then again, there is the hatch editor. You can assign a thickness to your cross hatches. You can also assign a different color but I am not sure if it will take on the style of that new color.
The Hatch Editor is not in the ribbon but can be found with the command search tool. This might be a workaround if you always want your hatch to be thicker than standards dictate.