  In this paper, we study the Johnson homomorphisms of basis-conjugating
automorphism groups of free groups. We construct obstructions for the
surjectivity of the Johnson homomorphisms. By using it, we determine its
cokernels of degree up to four, and give further observations for degree
greater than four.
  As applications, we give the affirmative answer for the Andreadakis problem
for degree four. We show that the cup product map of the first cohomology
groups of the basis-conjugating automorphism group of a free group into the
second cohomology group is surjective. Finally, we calculate the twisted first
cohomology groups of the braid-permutation automorphism groups of a free group.
