Although the Spanish Inquisition was partly an attempt to force people to adhere to the beliefs and practices of the Catholic Church, it was not really under the control of the Church. Instead, it was run autonomously by Ferdinand and Isabella, the King and Queen of Spain.
The main purpose of the Spanish Inquisition was to root out people who were pretending to be Christian but were secretly Jewish or Muslim. The Spanish monarchs felt that this needed to be done for a number of reasons. As the link below says:
Apparently, the monarchs acted from several motivations: They wanted to save souls from condemnation, to promote religious conformity, to centralize their political control, to appease public prejudices, and also to raise funds.
The Pope gave permission for the Inquisition to begin and proceed, but was not in control of it. Instead, it was the King and Queen, heavily influenced by the Dominican order, who were in control.
Therefore, the role of the Catholic Church was limited. The Inquisition was carried out by priests. However, they were not acting under the direction of the Church hierarchy but rather under the direction of the Spanish crown.