I will write for you in summary what `Umar bin Sa`d ibn abi Waqqas's role was in the death of Husayn according to the book "al-Mihan" by abu al-`Arab:
When Muslim bin `Aqeel was arrested and before he was killed, he asked them to allow him to make a Wasiyyah, he looked at the faces of the people and saw `Umar bin Sa`d ibn abi Waqqas, he decided to offer his will to him and told him: "I see you are the only man from Quraysh here, come closer so I may tell you something." So he did and he told him: "If you wish to be from the masters of Quraysh for as long as Quraysh exists, then go to Husayn and his companions who are 90 individuals, men and women, and ask them to turn back and tell them what happened to me."
Muslim was killed so `Umar bin Sa`d told the man in charge that he heard something important, so he told him to keep the secrets of his cousin and not say, but `Umar insisted that he must know because it is a matter of great importance then told him about Husayn. The man said: "Now that you have said this, no one is to go and fight him other than you!"
After Husayn and his companions fought a group from the filth of `Iraq near the water, they won and were able to rest and drink. `Ubaydullah ibn Ziyad had meanwhile sent `Umar bin Sa`d with an army, and when they met Husayn and his companions, Husayn stood up and said to Ibn Sa`d: "I give you one of three choices: Either you let me go back, or you send me to Yazeed and I place my hand in his hand and he shall know how to judge me, or you send me to the Turks so I may fight until death."
`Umar ibn Sa`d sent this request to Ibn Ziyad who in turn decided to send him to Yazeed. A filthy man called Shahr ibn Hawshab (Not the popular narrator) told Ibn Ziyad that he must not let him go, he must pass his own judgement on him and Husayn must accept.
Husayn of course rejected this, and `Umar ibn Sa`d intentionally wasted time so as to not fight Husayn. Ibn Ziyad told Shahr: "If `Umar fights Husayn then fight, if he does not then kill him and take his place."
Thirty Koufans in `Umar ibn Sa`d's army became angry at the situation and defected to Husayn's side. A soldier from Ibn Sa`d's army went and killed a family member of Husayn, Husayn then retaliated by killing the man so a fight broke out and what happened happened.
^ That's what I found about `Umar ibn Sa`d in this book. We need to look at all the other books of course to get the most accurate story.