Yes they do! And this ability of methyl groups to be copied at each round of DNA replication is what makes DNA methylation "epigenetics": the information carried by these methyl groups is faithfully and autonomously transmitted from the mother cell to its daughter cells. There is a specialized enzymatic complex, the DNA methylltransferase 1 (DNMT1), which reads the methyl groups of the old DNA strand during the replication process, and copies them at the exact same position on the newly synthetized strand. Like a carbon copy.
answered by: Deborah Bourc'his