Is it permissible for the head of the support services department to assign a worker (carpenter, plumber, electrician, etc.) to perform his work in a place where women are present, and does he bear sin for that?

Mixing [of genders] is not absolutely forbidden; rather, it has details. It is forbidden if it involves private seclusion (khalwa) with a non-mahram woman, looking with lust, the woman being immodest, or physical contact. It is permissible if there is a legitimate need, while observing the rules of Sharia, such as a woman going out for congregational prayers. Therefore, there is no harm in workers being present in a place due to need, provided they commit to lowering their gaze, avoiding unlawful seclusion, not speaking unnecessarily, and choosing the most righteous among them.