You can combine this function for finding the mode with aggregate.
Mode <- function(x) {
ux <- unique(x)
ux[which.max(tabulate(match(x, ux)))]
}
aggregate(someuser ~ groupname, Data, Mode)
groupname someuser
1 a x
2 b x
3 c x
Note that in the event of a tie, it will only return the first value.