Refer to figure.
At initial contact, when establishing communications, an aircraft should use the full call sign of both the aircraft and the aeronautical station. An aircraft shall use its abbreviated call sign only after it has been addressed in this manner by the aeronautical station. Therefore, the pilot should have used the following call sign:

• "A6-ETT" = Alpha Six Echo Tango Tango. And NOT Alpha Six Tango Tango. Furthermote, the latter is also not the correct abbreviated call sign.

The call sign "A6-ETT" consists of the characters corresponding to the registration marking of the aircraft. After satisfactory communication has been established, and provided that no confusion is likely to occur, the call sign may be abbreviated as follows:

• the first and at least the last two characters of the aircraft registration => A-TT "Alpha Tango Tango".

Based on the above, we can confidently confirm that Chester radar abbreviated the call sign correctly.