Problems with do add_row and group_by. Need the group_by variable name in add_row

Problem: I want to add_row using dplyr/tibble. I want to group the data by A in my example, and then add_row that contains the group name A and then a value for B.

The problem I am facing is trying to add the Group_by variable A in the column under A. No matter what I try it always comes back with an error or NA as the value in that column.

1. Reproducible example:

   example <- data.frame(A = sample(letters[1:3],10,replace = TRUE), 
                 B = sample(letters[24:26],10,replace = TRUE),
                 C = sample(1:3,10,replace = TRUE))
   

2. Output of example data:

         A B C
      1  c y 2
      2  b x 3
      3  c y 1
      4  b y 1
      5  c z 1
      6  a x 1
      7  b x 1
      8  c z 2
      9  a y 3
      10 c y 1
   

3. Code I want to run.

   answer <- example %>%
     mutate(A = as.character(A),
            B = as.character(B)) %>%
     group_by(A) %>%
     do(add_row(., 
                B = "ADDED",
                C = "ADDED"))
   

4. Output of the data:

           A     B     C
     1     a     x     1
     2     a     y     3
     3  <NA> ADDED ADDED
     4     b     x     3
     5     b     y     1
     6     b     x     1
     7  <NA> ADDED ADDED
     8     c     y     2
     9     c     y     1
    10     c     z     1
    11     c     z     2
    12     c     y     1
    13  <NA> ADDED ADDED
   

5. So in the output of the data, where there is NA I would like it to say the group name (a,b, or c)

I have tried putting just the group variable name there, it does not work throws an error.

Thanks!