Here's some basic data about Pz.Jg.Abt. 510 based on various NARA and BA/MA documents:
10.12.1944 OKH/GenStdH/Org.Abt. issues ordes for creation of 10. Radf.Jg.Brig.. The brigade initially is authorised to have a Pz.Jg.Kp. 1510 equipped with 14 Jagdpz. 38. This company is formed a few days later in Milowitz using Personaleinheit (P.E.) 2a.
15.12.1944 OKH/GenStdH/Op.Abt. orders transport of Pz.Jg.Kp. 1510 'Festung Loetzen'. There it is to relief Pz.Jg.Kp. 1106, which in turn is ordered to move to Sensburg for incorporation into 10. Radf.Jg.Brig.
16.12.1944 14 Jagdpz. 38 are shipped from a H.Za. to P.E. 2a, i.e. Pz.Jg.Kp. 1510
17.12.1944 Pz.Jg.Kp. 1510 leaves Milowitz
28.12.1944 OKH/GenStdH/Org.Abt. issues ordes for creation of Pz.Jg.Abt. 510 for the 10. Radf.Jg.Brig. in Milowitz:
Stad & Stabskp. 510 are to be formed using remnants of the former Stab & Stabskp./Pz.Jg.Abt. 125
1./510 is to be formed by renaming Pz.Jg.Kp. 1106 (already with the brigade)
2./510 is to be formed using P.E. 10a
3./510 is to be formed using P.E. 5c
Combat readyness for the unit is to be achieved until 20.1.1945
27.12.1944 Pz.Jg.Kp. 1510 is tactically attached to the 73. Inf.Div. (the divisions own Pz.Jg.Kp. 1173 is still training in Milowitz at the time). The company later is encircled together with the 73. Inf.Div. in 'Festung Thorn' and all Jagdpz. 38 are lost during the breakout. Surviving personnel arrives back in Milowitz on 20.2.1945. Here the men stay until the end of the war. Refit with new tanks does not happen again.
The 1./Pz.Jg.Abt. 510 (former Pz.Jg.Kp. 1106) actually joined the 10. Radf.Jg.Brig.. The last known strength report of the brigade from 26.2.1945 shows the company with 5 Jagdpz. 38 left. It also fielded a Marder III (7,5 cm) which probably came from the 4.(s)/Aufkl.Lehr-Abt. which had been incorporated into the brigade as I. Bataillon. Later the remnants of the brigade are incorporated into the 28.Jg.Div. in mid/late March 1945 and on 15.4.1945 it is officially disbanded.
26.1.1945 OKH/GenStdH/Org.Abt. assigns 2. and 3./Pz.Jg.Abt. 510 - still in Milowitz - to Pz.Jg.Brig. 104.
29.1.1945 14 Jagdpz. 38 and 1 Bergepz. 38 each are shipped from a H.Za. to 2. and 3./Pz.Jg.Abt. 510
29.1.1945 OKH/ObdE/AHA renames 2. and 3./Pz.Jg.Abt. 510 to 1. and 2./Pz.Jagd-Abt.(gp.) 5
1.2.1945 Previous order cancelled, 2. and 3./Pz.Jg.Abt. 510 renamed to 1. and 2./Pz.Jagd-Abt.(gp.) 3
9.2.1945 OKH/GenStdH/Op.Abt. orders transport of Pz.Jagd-Abt.(gp.) 3 to Sagan in the area of H.Gr. Mitte
17.2.1945 OKH/ObdE/AHA orders creation of Pz.Jg.Abt. 'Glienecke' in Berlin-Gross-Glienecke for the defense of Berlin using the now 'jobless' Stab and Stabskp./Pz.Jg.Abt. 510, P.E. 33b, 13c and 1b. Unit is to be combat ready by 21.2.1945.
20.2.1945 OKH/ObdE/AHA orders creation of Pz.Abt. 'Jüterbog' for the new Pz.Div. 'Jüterbog' using Pz.Jg.Abt. 'Glienecke'. As a 4. Kp. the P.E. 20a is added. Pz.Abt. 'Jüterbog' received the following tanks:
21.2.1945 10 Pz. IV/70(V) shipped from a H.Za. (for the 4. Kp.)
25.2.1945 30 StuG III shipped from a H.Za.
2.3.1945 1 Pz.Kpfw. IV shipped from a H.Za.
1.3.1945 OKH/GenStdH/Org.Abt. orders incorporation of Pz.Div. 'Jüterbog' into 16. Pz.Div.. The former III./Pz.Rgt. 2 of that division is disbanded and replaced by Pz.Abt. 'Jüterbog' as new III./Pz.Rgt. 2.
The order does'nt say it, but from other 16. Pz.Div's documents it appears as if the 4.(Pz. IV/70) Kp. was incorporated into the I./Pz.Rgt. 2
So as you can see the story of Pz.Jg.Abt. 510 is indeed somewhat complicated, and believe me it took me some time to figure it out!