Compare one col with another col in another df containing multiple entries

Question

We are trying to find if the char in A$symbol matches any of the char in B$symbol. Result should be 3 df+ one with matches, one with only in A, one with only in B.

Data example:

### task dfs

A = data.frame(
  symbol_A = c("ABC", "ABB", "ACC", "BCG", "AAG"),
  id_A = c("1", "2", "3", "4", "5"))

B = data.frame(
  symbol_B = c("XXC", "XCT ABB", "TTG WHO ACC", "AAG", "TTR, YHD"),
  id_B = c("ab", "dy", "hu", "uh", "yz"))


### expected solution 

solution_overlaps <- data.frame(
  symbol_A = c("ABB", "ACC", "AAG"),
  symbol_B = c("XCT ABB", "TTG WHO ACC", 'AAG'),
  id_x = c("2", "3", "5"),
  id_y = c('dy', 'hu', "uh" )
)

solution_only_in_A <- data.frame(
  symbol_A = c("ABC", "BCG"),
  id_A = c('ab', '4')
)

solution_only_in_B <- data.frame(
  symbol_B = c("XXC", "TTR, YHD"),
  id_B = c('ab', "yz")
)

Thanks a lot for helping, hope this is relevant for others as well! (as always, dplyr appreciated!)

Sebastian