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JELLY 10 9 bytes

f⁴L
ŒgÇ€Ṁ

Explanation:

f⁴L
f⁴      -Filter by the character argument.
  L     -Return Length of filtered String.

ŒgÇ€»/
Œg      -Group string by runs of characters.
  ǀ    -Run above function on each group.
    Ṁ   -Return the largest in the list.

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