State running Austrian school, having sacked Head (5,8)

'sacked head' is the definition.
The answer and definition can be both related to communication as well as being singular nouns.
Perhaps they are linked in a way I don't understand?

'state running austrian school' is the wordplay.
I cannot really understand how this works, but
'state' could be 'first' (first is a kind of state) and 'first' is found within the answer.
'school' could be 'language' (language school is a kind of school) and 'language' is found in the answer.
No letters remain.
This may be the basis of the clue (or it may be nonsense).

'having' is the link.

Can you help me to learn more?