It's not a problem that's unique to the Bundeswehr. The US military have problems with far right nutters in their ranks and in the NZDF we have one individual who is subject to disciplinary action for passing on NZDF information to a far right group.
As per May 7th 2021 military justice has
about 50 disciplinary cases open with regard to KSK, of which about half are at a stage ready to be tried. About 30 of these are explicitly for rightwing extremism, the rest is mostly for (related) ammunition disappearances and -mishandling.
There is also an ongoing investigation into procurement processes at KSK which will result in more disciplinary cases but is not counted among the above.
The internal intelligence service MAD since beginning investigations into and observations of the group in 2017 has investigated "about 50" cases of extremism suspicion within KSK. 7 of these cases were cleared ("green"), 24 cases have ended (suspect sentenced, dismissed, retired or transfered out of KSK), about 20 cases are still active. One case was switched from "suspect" to "proven" ("orange")
after the dissolution of 2nd Company.
MAD has explicitly not found a group intending to remove the free-democratic basic order of Germany or elements of it (as in a group that could be outlawed and members punished due to their support). It has however found a "mesh" (they're trying to avoid the word network) of mutual acquaintances of various levels between suspects that is based upon a "common mentality" and is being further investigated.
Out of 66 soldiers that were part of 2nd Company at its dissolution 61 were transfered internally and remain within either KSK or in the training command or recruiting teams separated from KSK since then.
Brigadier General Kreitmayer, the head of KSK, in September will be transferred to a post in the Armed Forces Office that is used for "problematic" cases that the Bundeswehr wants to keep out of the public light. For some scale, that post is currently held by Brigadier General Georg Klein, who in 2009 ordered a controversial airstrike in which about 100 civilians died.
Kreitmayer's replacement will be Brigadier General Ansgar Meyer, the current commander of TAAC-N - i.e. the German Forces in Afghanistan - who will need a new job anyway once the withdrawal concludes. He will additionally now have to report to a - so far unnamed - new "Director of Special Forces" at the Army Command who will be set up to "coordinate and oversee" the special forces of Army, Navy and Air Force (of which KSK is about 70%).
Major General Hannemann, head of the Rapid Forces Division (and hence the commander above Kreitmayer), will be laterally transferred to become deputy commander of the German-Dutch Corps. Hannemann came under some scrutiny for lack of oversight in logistics processes, but is not accused of anything in this regard. The transfer and replacement in his case is supposedly unrelated to the investigations.
