In terms of Gertrude in Shakespeare's Hamlet, and England's Queen Elizabeth I, there is a clear distinction between the two in terms of their leadership capabilities.
Elizabeth I was very forthright, like her father—Henry VIII. She was very adroit in the political dance with would-be suitors from "super powers" that wanted to form an alliance with England. Known as "The Virgin Queen," she kept her marriageability "on the table" until she was no longer in childbearing years. Elizabeth used this as a carrot, making "vague promises" to the leaders of other countries, but it seems that she really had no desire to marry. Her advisors pushed her to do so for an heir, for many years, but she refused.
In terms of leadership, Elizabeth was able to address the major concerns that faced England for most of her reign. Sir Francis Drake, leader of her navy, harassed the Spanish ships a great deal mostly because they were returning with great riches from the New World. Each ship captured delivered large amounts of wealth to the depleted English treasury. The Spanish complained profusely; but there is some question as to whether these were sanctioned attacks or not. Elizabeth's fancy footwork kept the Spanish at bay for a long time. Eventually, the Spanish Armada, once known as the most powerful navy in the world, sailed for England in 1588. While her men waited for the attack, Elizabeth was recorded as saying:
I know I have but the body of a weak and feeble woman; but I have the heart of a king, and of a king of England.
The Armada was defeated, in the midst of an opportune storm.
Elizabeth was harsh when the situation required it, and she did not make exceptions for friends or family. This was not an easy thing for her. When a favorite, the Earl of Essex, and later her cousin, Mary, Queen of Scots, were convicted of treason in trying to usurp her position on the throne, Elizabeth sadly signed the documents required for their deaths.
Elizabeth I inspired those who followed her. She took her sovereignty very seriously. In all, she loved her country and her people. She said:
Though God hath raised me high, yet this I count the glory of my crown, that I have reigned with your loves…
Gertrude, in Hamlet, has little power, and therefore, virtually no ability to lead. She bows to the superior position of Claudius, her second husband. Although Claudius lets on that his actions towards Hamlet must be handled carefully so as not to upset his "doting" mother, Claudius is clearly in charge. Gertrude is included in plans to spy on Hamlet, but Claudius secretly makes his "darker" plans, knowing how Gertrude loves her son. This is really the only power she has: to intervene for her son, and this "power" may only exist because Hamlet is well-loved by the Danish people.
The only other time Gertrude seems to have any control is when Ophelia's mind begins to fail. Perhaps Claudius does not know what to do, or perhaps because Gertrude has a special affection for Ophelia, Gertrude intercedes with Claudius for his help, at which time he sends Horatio after Ophelia to watch her and keep her safe.
Hamlet does not call for a strong queen. Her place is beside her husband, which may be why she marries Claudius after Old Hamlet's death: for protection of herself and her son. Marriage guarantees her place, and Hamlet's, in Denmark. This may be, in reality, the only "power" she has. She is wife to the King—she is "queen" in name only. She cannot lead at all.