$\mathrm{S}-\mathrm{G}+{\mathrm{S}}^1\to \mathrm{S}+{\mathrm{S}}^1-\mathrm{G}$ enzyme catalysing the above reaction belongs to the class namely

Transferases: Enzymes catalysing a transfer of a group, G (other than
hydrogen) between a pair of substrate S and S’ e.g.,
S - G + S’ ---- S + S’ - G