In Shakespeare's Hamlet, the Ghost has two very different desires in mind when it comes to how his son Hamlet will avenge Old Hamlet's murder.
It is in the fifth scene of Act One that Old Hamlet gives his son direction regarding how he should proceed in punishing his brother Claudius who not only wears Old Hamlet's crown, but also sleeps with Gertrude in the bonds of incestuous marriage. He also instructs Hamlet as to how he should treat his mother.
If thou hast nature in thee, bear it not.
Let not the royal bed of Denmark be
A couch for luxury and damned incest.
But, howsoever thou pursuest this act,
Taint not thy mind, nor let thy soul contrive
Against thy mother aught. Leave her to heaven,
And to those thorns that in her bosom lodge
To prick and sting her. (86-93)
It is noteworthy that Elizabethans believed that when a man and woman married, they became as one. It was understood that one could no longer tell where the husband's physicality ended and his wife's began because they were joined so closely in flesh and in spirit. With this said, Elizabethans also believed that if one died, the essence of the deceased spouse lingered on, still living in the spouse that survived. Claudius is Old Hamlet's brother. With the conviction that something of Old Hamlet still lived in Gertrude, Old Hamlet and Hamlet refer to the incestuous marriage of which Gertrude is a part. This means that when she and her new husband sleep together, the essence of Old Hamlet within Gertrude is also having sex with his brother Claudius. This certainty would have greatly offended Shakespeare's audience. Not only would its members be horrified by Gertrude's behavior in marrying her brother-in-law, but also (and even more) they would be disgusted by Claudius' marriage to his brother's widow.
The Ghost tells Hamlet that as far as he is concerned, Hamlet should not let the marriage continue is such a state—making the king's bed a mockery of matrimony with such lecherous behavior. The only way this can take place is if Claudius is killed. However, killing a king is also a mortal sin. It is because of his father's wish and his fear of losing his soul that Hamlet spends so much time in the play trying to prove that Claudius is guilty. Then later, even when he has done so, Hamlet refrains from killing the King because he is praying. (Hamlet does not want Claudius to go to heaven having confessed all of his sins when Hamlet's father was never afforded the same opportunity and now must suffer in purgatory.)
Old Hamlet has far different wishes where Gertrude is concerned. The Ghost instructs Hamlet to make no move against his mother. Instead, he tells his son to let heaven deal with her (in whatever manner God may choose), and let her live on with the stabbing of her conscience to feed the guilt she experiences over what she has done.