Mu`awiya who ostensibly wanted to avenge the murder of...
Mu`awiya who ostensibly wanted to avenge the murder of Uthman was in reality keen to strengthen his own rule as well as that of his descendants. His efforts were directed towards strengthening his rule in Syria in the first instance. Then he wanted to expand his kingdom by grabbing other countries and eventually to hecome the despotic ruler of all the Islamic territories. He did not pay any heed to Uthman either during his lifetime or after his death.
All that he wanted was that Uthman might make him powerful day after day so that he might eventually achieve his final goal. Thus he desired that Uthman should allow him maximum freedom of action and should become a shield for him, so that he might achieve his objectives. Even when Uthman was killed Mu`awiya did not care for his death. He only wanted to avail of the opportunity to claim to be the caliph's heir, and to get rid of the new caliph.
What did he do with the murderers of Uthman when he himself became the sole ruler of the Islamic territories? Had he really been grieved on account of the murder of Uthman he could then locate and kill every one of his murderers. However, he completely forgot Uthman's murder and revenging him, although it was on this very pretext that he had revolted against the new caliph and thus became the cause of bloodshed of hundreds of thousands of the Muslims.
Furthermore, he could send a large army from Syria to defend Uthman when his house had been besieged by the insurgents. He was the permanent Governor of Syria and Uthman had allowed him maximum freedom. He could do what he liked and there was none who could call him to account. Thus he could send a large Syrian army to Madina before as well as after the house of Uthman was besieged. In fact he could also advise Uthman not to be adamant to public opinion.
However, he did not do any of these things, because he was keen to grab the caliphate after Uthman and could not think of anything else. From the very day on which Uthman called a conference of his confidants, which was also attended by Mu`awiya, and which ended without taking any decision, Mu`awiya decided finally to seize the caliphate, because he became sure that Uthman would be killed.
As Syria was in his hands and the inhabitants of that region were obedient to him, he realized that if Uthman was killed he would acquire the best weapon (viz. vengeance of the murder of Uthman) to achieve his purpose.