I have done this with dummy data.

Lets say we have we have regions A (groupA) and regions B (groupB).

groupA <- data.table(
    chr = rep("chr1", 5),
    start = seq(0, 1000, 100),
    end = seq(50, 1050, 100))
setkey(groupA, chr, start, end)

groupB <- data.table(
    chr = rep("chr1", 5),
    start = seq(25, 1025, 100),
    end = seq(75, 1075, 100))
setkey(groupB, chr, start, end)

Check:

1. If your datasets are data.table class(groupA), if not do setDT(groupA)
2. If keys are chr start end, if not do setkey(groupA, chr, start, end)

# Find overlaps
over <- foverlaps(groupA, groupB, nomatch = 0)
# Extract exact regions
over2 <- data.table(
    chr = over$chr,
    start = over[, ifelse(start > i.start, start, i.start)],
    end = over[, ifelse(end < i.end, end, i.end)])

The findOverlaps() method is not the right tool for the job. Instead, you want the set intersection which can be obtained with intersect() on two GRanges objects.

# create two GRanges objects for testing
gr1 = GRanges(seqnames='chr1',ranges=IRanges(start=seq(1,1001,100),width=50))
gr2 = GRanges(seqnames='chr1',ranges=IRanges(start=seq(25,1025,100),width=50))

# and "intersect" them, finding the set of regions
# formed by the overlaps of gr1 and gr2
gr3 = intersect(gr1,gr2)

# and the result
gr3

GRanges object with 11 ranges and 0 metadata columns:
       seqnames       ranges strand
          <Rle>    <IRanges>  <Rle>
   [1]     chr1 [  25,   50]      *
   [2]     chr1 [ 125,  150]      *
   [3]     chr1 [ 225,  250]      *
   [4]     chr1 [ 325,  350]      *
   [5]     chr1 [ 425,  450]      *
   [6]     chr1 [ 525,  550]      *
   [7]     chr1 [ 625,  650]      *
   [8]     chr1 [ 725,  750]      *
   [9]     chr1 [ 825,  850]      *
  [10]     chr1 [ 925,  950]      *
  [11]     chr1 [1025, 1050]      *
  -------
  seqinfo: 1 sequence from an unspecified genome; no seqlengths