.......Would you mind to explain who is in charge of the aircraft for the whole trip?
Normally, in my experience flying for two International airlines, one Captain is designated as being " In Command " and the other will be subordinate legally, but it doesn't stop the arguments between them of course, or the unspoken dominating personality of one over the other. As has been said before, Human Nature Rules, OK ?
That is why many prefer the augmenting crew to be off the flight deck when not actually required - just easier, sometimes.