No, the ghost instructs Hamlet to seek revenge on his uncle, King Claudius, for murdering him and usurping the throne. The ghost does not mention punishing Gertrude, Hamlet's mother.
In Shakespeare's "Hamlet", The Ghost demands Hamlet kill Claudius, the brother who betrayed, murdered and stole his throne and wife. However, this murderous vengeance does not extend to his offending wife. He tells Hamlet not to seek vengeance on her and to just focus on dealing with the usurper.
Hamlet is talking to the Ghost, which Gertrude can't see, which looks bizarre to her. For the first time, Gertrude sees Hamlet as seriously mentally ill, not just disturbed. Ironically, it's Gertrude whose perceptions are faulty in this scene.
the ghost of his father
He reveals that Claudius murdered him. He doesn't tell Hamlet anything about Gertrude he didn't already know.
Hamlet sees his father's ghost, but the ghost is invisible to Gertrude.
The ghost tells Hamlet the secret of his father's death. Hamlet's father was not killed by a serpent, but was murdered with poison. He also told hamlet that it was his uncle who murdered his father, and that he also seduced Gertrude, his mother. He orders him to revenge his murder against Claudius but to leave Gertrude alone.
Gertrude. I suppose he may have told Hamlet not to take revenge on Gertrude because the ghost still has affection for her, or because it would be particularly evil for a son to kill his mother, or because the ghost knows that Gertrude was not privy to Claudius's murder, and the murder is what he wants revenge for. Or some combination of these.
The ghost warns hamlet not to do anything to his mother, but to "leave her to heaven".
The only characters in the play Hamlet who are related to Hamlet are the ghost of his father King Hamlet, his mother Gertrude and his uncle Claudius. None of the rest are related to him.
The ghost of Old Hamlet warns Hamlet to not scold Gertrude. Hamlet was warned earlier "leaver her to heaven" (V v) and was warned later in the third act. The ghost warned Hamlet probably for one of two reasons, he still loved her or trying to change Gertrude would not work and who tried would go mad.
He did not. The ghost is only heard by Hamlet but Bernardo, Marcellus and Horatio see it. This is a clue to us that the ghost is not just a figment of Hamlet's imagination. Gertrude, however, cannot see or hear the ghost even when it is present in the same room. This suggests that the ghost can appear to anyone if that suits its purpose; it is important to be seen by the guards so they will bring it to Hamlet's attention, but also important that he not be seen to be involved by his widow. But the real reason, as ever in a Shakespeare play, is dramatic. The scenes where the ghost appears to the guards is a dramatic scene good for opening the play, and the fact that they have seen the ghost allows for a number of wonderful scenes: Horatio's description of the ghost in 1,2, the attempts by Horatio to hold Hamlet back in 1,3, and Hamlet's insistence that they swear an oath of secrecy. All of this would be lost if the Ghost was visible only to Hamlet. But in Act 3, the scene is between Hamlet and Gertrude, and the ghost is a minor distraction. Having him appear to Gertrude would necessitate some kind of confrontation between them, and the scene would become one between Gertrude and the Ghost and Hamlet would be superfluous. Since such a scene is not necessary to the plot (Hamlet will convince Gertrude of Claudius's guilt without the help of the ghost), Shakespeare wisely avoided it.
Hamlet is talking to the ghost of his father. Unfortunately for him, on this occasion, the ghost is visible and audible to Hamlet but not to anyone else, particularly Gertrude, who is sitting right beside him. She sees Hamlet talking to empty space, and asks him why he is looking at and talking to nothing. She is convinced that he's nuts: "This is the very coinage of your brain."