The Wikipedia article on Kampfgeschwader 6 claims that it was only planned to re-designate the bomber formation into a fighter formation ("Jäger" – J), but that these plans were given up due to 'unlucky events', like 'end of the war'.
In September 1944 the crews left their bombers behind, moved to Prague and started re-training with fighters which as of December consisted of:
Group I: 19 Bf 109 G-6/G-10; Group II: 19 Bf 109 G-6; Group III 10 Fw 190 A-8.
Only Group III then received Me262 in late January and was reformed into the "Gefechtsverband Hogeback", while the other groups were given up as of April 9. The Hogeback Group operated from Prag-Rusin airfield. In April and until May 5 they flew missions from there.
— Lexikon der Wehrmacht: Kampfgeschwader 6, Kampfgeschwader (Jagd) 6, Gefechtsverband Hogeback
This is consistent with the Wikipedia description:
Luftwaffe airfields identified as jet bases were frequently bombed by medium bombers, and Allied fighters patrolled over the fields to attack jets trying to land. The Luftwaffe countered by installing extensive Flak alleys of anti-aircraft guns along the approach lines to protect the Me 262s from the ground—and by providing top cover during the jets' takeoff and landing with the most advanced Luftwaffe single-engined fighters, the Focke-Wulf Fw 190D and (just becoming available in 1945) Focke-Wulf Ta 152H
Group III/KG(J)6 seems to have operated a few times Fw 190A and Me 262A from Prag-Rusin, but for a time Group II was at the same airfield with Bf109.
— Luftwaffe units: Kampfgeschwader 6
A photo of Fw 190 and Me 262 in the same hangar, albeit for Jagdgeschwader 7 in Stendal:
— Word War 2 Database: Personnel of the US 5th Armored Division examining a captured German Me 262 Schwalbe of Jagdgeschwader 7 at Stendal, Germany, 15 Apr 1945. Note FW-190 at right.
Note that the German Wikipedia entry on 'Me 262' does not list KG6(J) or Gefechtsverband Hogeback at all as any unit operating Me 262, and the entry for KG6 lists as last major operation nothing near Prague. (Based on: — Wolfgang Dierich: "Die Verbände der Luftwaffe 1935–1945. Gliederungen und Kurzchroniken ein Dokument", Verlag Heinz Nickel: Zweibrücken, 1993, (p703).
The following author only lists the demise of Groups I and II, which were to be transferred to Graz to be converted into a paratrooper formation.
— Georg Tessin: "Verbände und Truppen der deutschen Wehrmacht und Waffen-SS im Zweiten Weltkrieg 1939–1945. Band 14, Fliegende Verbände" Biblio-Verlag 1998, p373.