Let me describe you a hypothetical that motivates the subject question; a German citizen has been sent on an official government business overseas (let's say an ambassador outside of the EU). On this mission, he acts in manner than would be clearly illegal by legal standards of the EU against another EU citizen, inflicting him with provable damages. Specifically, let's say he asks the host country in writing to act against said EU citizen, for example to strip him of citizenship which they (the host country) comply. Are there grounds for a lawsuit against said ambassador or the government of Germany in the courts of Germany with reasonable expectation of success?
Does Germany excercise extraterritorial jurisdiction over its citizens?
Yes, the German Criminal Code (StGB) will be applied if any of the conditions defined in §§ 4-7 StGB (German Criminal Code) are fullfilled.
This assumes that the charges made are punishable as a crime defined in the StGB.
Specifically, let's say he asks the host country in writing to act against said EU citizen, for example to strip him of citizenship which they (the host country) comply.
This is probably not covered in the StGB.
He was encouraged to do so on a press conference. I don't think malice intent was there,...
Since intent to commit a crime is an important condition that must be fulfilled, it is unlikely that any charges would succeed.
Here you would have to find a condition inside the StGB where negligence would apply.
In your main text you state: he asks the host country in writing to act against said EU citizen, but later write: encouraged to do so on a press conference.
Such contradictory statements would not be received well by a court in any jurisdiction.
If the press conference statement only meantioned that the person in question is an EU-Citizen in a country (that does not allow dual citizenship) where that person is also a citizen, the person in question must supply proof (acceptable to the court) that it was done with the intention to cause damage to the person in question.
It is unlikely that any court would hold anybody responsible for accidentally revealing an illegal behaviour of the person in question.