  We investigate the pair creation of magnetically charged black holes on a
cosmic string in the presence of a background magnetic field. The string may
either break or fray to produce a pair of accelerating black holes described by
Ernst metric. By using the instanton action we obtain the rate of such
production. For large values of background magnetic field the production of
large black holes is probable. Comparing our results with the case of black
hole pair creation in magnetic field with no string, we show that
fraying/breaking of the cosmic string can substantially enhance the production
rate. We also obtain the result of monopole-antimonopole pair production on
cosmic string in an external magnetic field, using the WKB approximation and
compare it with the black hole results. We also provide a heuristic study of
black hole pair creation on cosmic string in the presence of a magnetic field
in the de Sitter background, although the analog of Ernst metric with a
cosmological constant is lacking. Like a cosmic string and/or a background
magnetic field, a positive cosmological constant increases the production rate.
