My main question is in the title:
As the Logicians are not very popular among the Adeptus Mechanicus I was wondering which Mechanicus sects that would activily go out and route them out for purging?

Secondly:
And how would these Mechanicus sects interact with the Inquistion and Ministorum, as they don't like the Logicians either? Could it be three different groups (Mechanicus, Inquisition, Ministorum) hunting the same Logician cell without any of them knowing about the other groups' effort?

"Actively" only leaves room for thus groups who are eiter some kind of secret service or militäry in nature.

The Light of Sollex is military in outlook, as well as any "Secutor" or detachments of the Skitarii (which are not much definied till now).

Any Explorator ship might in truce be a vessel designated to the hunt of these or other Hereteks

The Divisio Investigatus, the Techsorcists, the vassals of the Lords Dragon, the Dragon Secutorii, the Auxilla Myrmidon…


In regard to the cooperation with an Inquisitor: they either would form joint teams for a whiler or one might place his servants under the command of another.

Depending on what the logicians are making, the AdMech may not want the Inquisition or ministorum to know about it.
As far as i understand (i may be wrong) The inquisition have no real power on forgeworlds so the if the Admech see the logicians as a threat to their domain, they may want to exclude other agencies. May even go so far as to kill other agents (would deny it of course).

Of course, if some clue, rumour or evidence of heretek activity reached the ears of the cogboys, I think the list of those who might take an active interest grows quite quickly.

Ie, while perhaps only some groups expend resources actually looking for trouble, there will be many who are happy to join the fray once the active bullies have pointed out the schoolyard deviants…. eh, I mean Vile Heret(ic/ek)s.

And generally people like to point out where others are wrong most eagerly in areas they are most interested in. So I would assume that groups in the Ad Mech dabbling in bio-tech would be first in line to hunt those messing with genegrafting, while those tinkering with the advanced machine spirits of sentient machines such as Titans would be zealous in their persecution of those messing with Abonimable Intelligences and whatnot.

If you haven't already, read up on the Logicians in Disciples of the Dark Gods. It has some concrete examples for you.

The Holy Ordos’ wrath was terrible, and in conjunction with the
Mechanicus faction known as the Divine Light of Sollex,
they systematically hunted down and scourged the Logician
influence from Hive Tarsus on Scintilla, Fenksworld….

and

Inquisitor Srax-Rhame of the Ordo Hereticus has made it his
goal to confront and destroy the Logicians.

I am sure they are not the only ones