Welcome, Pichichi. Hope to see many thoughtful posts from you!
Roth was behind the attempted assassination of Michael at Tahoe. He sought to make it look like Pentangeli was behind the assassination. The attempt failed. But when Michael visited Roth in Miami, Roth was heartened that Michael, as he expected, blamed Frankie for the attempt. He promised to kill Frankie (“Frank Pentangeli is a dead man”), and even asked Roth's permission (“You don’t object?”), as you noted. Roth saw a silver lining in the cloud of the botched assassination: Michael would kill the obstreperous Frankie without his having to lift a finger or spend a dime! Heh-heh!

But, instead of killing Frankie, Michael dispatched him to settle his problems with the Rosato Brothers. Oh-oh! Roth knew Michael would never give a pass to a mortal enemy. So, it was certain that Michael didn't suspect Frankie in the Tahoe attack-—leading to the possibility that he did suspect Roth.

What to do? Roth ordered the Rosatos to kill Frankie at their meeting. That way (and I'm finally answering your question), he'd eliminate a Michael-ally before the two of them could cook up any mischief against him. To maintain his façade, Roth could always claim that he’d only done what Michael said he was going to do anyway. (Later, in Havana, Roth implied that Frankie’s assassination was tit-for-tat for Michael’s murder of Moe Green.)