For those who want to deny the complicity of the Church in the inquisitions, including the fact that heretics were executed by the State with the full understanding and consent of the Church, the following excerpts from the article at the Catholic Encyclopedia are worth reading (emphasis added):
Gregory IX may be regarded as having had no share either directly or indirectly in the death of condemned heretics. Not so the succeeding popes. In the Bull “Ad exstirpanda” (1252) Innocent IV says:
When those adjudged guilty of heresy have been given up to the civil power by the bishop or his representative, or the Inquisition, the podestà or chief magistrate of the city shall take them at once, and shall, within five days at the most, execute the laws made against them.
Moreover, he directs that this Bull and the corresponding regulations of Frederick II be entered in every city among the municipal statutes under pain of excommunication, which was also visited on those who failed to execute both the papal and the imperial decrees.
Nor could any doubt remain as to what civil regulations were meant, for the passages which ordered the burning of impenitent heretics were inserted in the papal decretals from the imperial constitutions “Commissis nobis” and “Inconsutibilem tunicam”…
The civil authorities, therefore, were enjoined by the popes, under pain of excommunication to execute the legal sentences that condemned impenitent heretics to the stake. It is to he noted that excommunication itself was no trifle, for, if the person excommunicated did not free himself from excommunication within a year, he was held by the legislation of that period to be a heretic, and incurred all the penalties that affected heresy. …
The Spanish Inquisition deserves neither the exaggerated praise nor the equally exaggerated vilification often bestowed on it… However, the predominant ecclesiastical nature of the institution can hardly be doubted. The Holy See sanctioned the institution, accorded to the grand inquisitor canonical installation and therewith judicial authority concerning matters of faith, while from the grand inquisitor jurisdiction passed down to the subsidiary tribunals under his control. Joseph de Maistre introduced the thesis that the Spanish Inquisition was mostly a civil tribunal; formerly, however, theologians never questioned its ecclesiastical nature. Only thus, indeed, can one explain how the Popes always admitted appeals from it to the Holy See, called to themselves entire trials and that at any stage of the proceedings, exempted whole classes of believers from its jurisdiction, intervened in the legislation, deposed grand inquisitors, and so on.
newadvent.org/cathen/08026a.htm