Does Islam permit killing the murderer or the adulterer, and what are the conditions for that, and how do we respond to those who say that God alone is the one who punishes?

It is not permissible to take a life except by right. This includes killing a murderer in just retribution (qisas), and killing a married fornicator, as the Prophet, peace and blessings be upon him, said: "The blood of a Muslim man who bears witness that there is no god but Allah and that I am the Messenger of Allah is not lawful to shed except for one of three reasons: a life for a life, a married fornicator, and one who abandons the religion and separates from the community." Killing a murderer is permissible if the killing was intentional and aggressive, and the guardians of the victim's blood (next of kin) do not pardon him. The retribution is carried out by the ruler. Killing a fornicator applies specifically to the married fornicator, and the prescribed punishment is not carried out against him except with the testimony of four upright witnesses. The statement of those who prohibit killing these individuals, arguing that only Allah punishes, needs to be clarified by emphasizing that their killing is a ruling from Allah Almighty. Its basis is not man-made law, but rather it is among the rights by which lives are taken, like the prescribed punishment of qisas that Allah has ordained.