How to add a column based on partial information from another column in R [duplicate]

I have an example data frame like the one below.

ID	File
1	11_213
2	13_256
3	11_223
4	12_389
5	14_456
6	12_345

And I want to add another column based on the number before the underscore in the "file" column to get a data frame that looks something like this.

ID	File	Group
1	11_213	GRP 11
2	13_256	GRP 13
3	11_223	GRP 11
4	12_389	GRP 12
5	14_456	GRP 14
6	12_345	GRP 12

I have tried using substrings within a series of if else statements like this:

df <- df %>%
mutate("Group" = ifelse(substring(File, 1,2) == "11", "GRP 11",
 ifelse(substring(File, 1,2) == "12", "GRP 12",
ifelse(substring(File, 1,2) == "13", "GRP 13", "GRP 14")))

And this works, but I realize this is a really inefficient way of doing this. Also, there are only so many ifelse statements you can do before you get an overflow error.

Is there a better way to do this? I think it might involve the grep or grepl functions, but I'm new to R and am not familiar with that. Any help would be great, thanks!