If partial string = "dup" is in Column A, insert a row below that cell.

try this:

Sub InsertAfterDup()
    Dim i As Long, M_ax As Long
Range("$A$2").Select
i = 0
M_ax = 0
While ActiveCell.Offset(i, 0) <> ""
  If ActiveCell.Offset(i, 1) > M_ax Then M_ax = ActiveCell.Offset(i, 1)
  If InStr(1, ActiveCell.Offset(i, 0), "dup", vbTextCompare) <> 0 Then
    i = i + 1
    ActiveCell.Offset(i, 0).EntireRow.Insert
    ActiveCell.Offset(i, 0) = ActiveCell.Offset(i - 1, 0) & "_max"
    ActiveCell.Offset(i, 1) = M_ax
    M_ax = 0
  End If
  i = i + 1
Wend
End Sub