Is capital punishment (Qisas) obligatory if some guardians of a victim kill the perpetrator without the permission of the remaining guardians?

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Capital punishment is not obligatory in this situation. This view is held by Abu Hanifa and is one of the opinions of Al-Shafi'i. The reasoning is that the killer is a participant in the right to exact retaliation, just as one who co-owns a slave girl and has relations with her is not subject to a specific penalty for that act. Furthermore, where some own a part of a specific entity, the prescribed punishment upon its complete disposal is not incumbent, similar to the principal entity itself. In the case where a group kills one person, Qisas is not imposed by killing only a part of the soul; rather, each individual is considered a killer of the whole. If we were to grant that Qisas is due for killing part of a soul, it requires participation in the act, which is not realized here.