You cannot add style to an SVG <g> element. Its only purpose is to group children. That means, too, that style attributes you give to it are given down to its children, so a fill="green" on the <g> means an automatic fill="green" on its child <rect> (as long as it has no own fill specification).
Your only option is to add a new <rect> to the SVG and place it accordingly to match the <g> children’s dimensions.