The Luftwaffe unit that took part in Operation Eiche (Oak), to rescue Mussolini in the Gran Sasso raid on 12th September 1943, was the Fallschirmjäger-Lehr-Bataillon, XI. Fliegerkorps. (Lehr means 'Training').
They were temporarily designated as I./FJR7 (I Bataillon, 7. Fallschirmjäger-Regiment) for the operation.
The Fallschirmjäger-Lehr-Bataillon was formed in July 1940 under the command of Major Burkhardt, and formed part of the Ramcke Brigade which was originally raised for the invasion of Malta in 1942.
The Brigade was sent to North Africa in August 1942 as a reinforcement for Rommel and entered the line at El Alamein. During the withdrawal of the Afrika Korps, the Brigade was abandoned by the high command since it had no organic transport. Rather than surrender, Ramcke led his troops out of the British trap and headed west, losing about 450 men in the process. They soon captured a British supply column which provided not only trucks but food, tobacco and other luxuries. about 600 later re-joined the Afrika Korps. The rump of the Brigade fought in the final battles in the defence of Tunisia. Many senior officers and some men escaped but most of the troops were captured in May 1943 when Tunisia finally fell to the Allies.
Most of the men in Burkhardt's battalion had been gradually flown out of Africa due to sickness. In fact, the battalion suffered far more from various illnesses than casualties inflicted by Allies. Hence, when the Ramcke Brigade was disbanded most of the men from the old Lehr Battalion found themselves in Germany and France. These men then became the cadre for a second formation of the Lehr Battalion, which was temporarily designated as I./FJR7 for the Mussolini rescue. Just after the Mussolini mission the batallion was then taken out of the 2.FJD and once more became the Lehr battalion. As such it fought at Anzio, where it wiped out a US Ranger battalion of nearly 800 men. Later the Lehr battalion was expanded to a regiment under the leadership of Knight's Cross holder Henry Herrmann.
Ramcke Brigade composition:
Battalion Kroh – formed from the 1st Battalion of Fallschirmjaeger Regiment2, just returned from Russia
Battalion Huebner – formed from the 2nd Battalion of the newly formedFallschirmjaeger Regiment 5
Battalion Burkhardt – formed from a demonstration/training (Lehr) Battalion
Battalion von der Heydte – a newly raised unit
Artillery Battalion – formed from the 2nd Battalion of 7th Flieger Division’sparachute artillery Regiment, commander was Fenski
Anti-tank Company, commander was Hasender
Signal Company
Pioneer Company, commander was Captain Cord Tietjen
I./FJR7 roll list at the time of the Gran Sasso raid (not all of them took part in the operation):
Batt Kommandeur - Major Otto Harald Mors, from 01.08.43 - 30.09.43
Adjutant - Oberleutnant Hans Joachim Kurth, later Leutnant Herold
Ordinance Offizier - Oberleutnant Alfred Romanskie, Oberleutnant Böger & Rolfs
N.-Zug - Leutnant Karl-Heinz Blumenthal
Hauptfw. - Feldwebel Herbert Ripke
Stabskompanie - Oberleutnant Hannes Weber, Kurth, Oberzahlm Becker, Leutnant Lungwitz, Oberzahlm Schmidt, Zalhm Both, Dr.Brunner, Dr.Schulze.
1.Kompanie - Oberleutnant Georg Freiherr von Berlepsch (Oberleutnant Weiss)
I.Zug - Leutnant Joswig - Feldwebel Rahden, (Feldwebel Bosshammer)
II.Zug - Feldwebel Eugen Abel - Feldwebel Alfred Asbach
III.Zug - Leutnant Hannes Weber+9.9.43, Leutnant Gradler - Feldwebel Matthias Heck
Kompanietrupp - Fw Bosshammer, Fw Vieth, Oberjäger Karrenbrock, Obj Irrgang, Gef Klein
2.Kompanie - Oberleutnant Heinz Georg Fischer (HFw Flöricke)
I.Zug - Leutnant Schmidt - Fw Otto Pawelcyk
II.Zug - Leutnant Otterbein - Fw Franz Kopp
III.Zug - Leutnant Wolter - Fw Erich Pliwischkies
3.Kompanie - Oberleutnant Karl Schulze (HFw Walter Schmidt, HFw Kleber)
I.Zug - Leutnant Hahn - Stabsfeldwebel A.Ehrmann
II.Zug - Leutnant Lücht - Leutnant Braun
III.Zug - Oberfeldwebel Schröder - Fw Treml
4.Kompanie - Hauptmann Lisowski (Fw Ernst Könnecke)
I.Zug - Leutnant Keller - Fw Lenhardt
II.Zug - Leutnant Opel - Fw Bunte (Fw Kempe)
III.Zug - Leutnant Ehrlich - Fw Wolk
https://www.feldgrau.com/ww2-german-fallschirmjäger-brigade-ramcke/
https://www.feldgrau.net/forum/viewtopic.php?t=27396
I./ FJR 7 - Sep 1943 (Gran Sasso). - Axis History Forum (this one is a detailed description of the raid with first hand accounts, including may from Skorzeny).
https://www.flamesofwar.com/Default.aspx?tabid=112&art_id=698&kb_cat_id=34