Do employers pay more when a former employee collects unemployment?

I’m talking about employers in the United States of America, where as I understand it, the Federal government has a program that all of the states go along with, where as employees are paid, some money contributes to an “unemployment insurance” program, and then if the employee is let go through no fault of their own, they can collect from the program while they are looking for new paid work.

I know that employers pay into the unemployment insurance (UI) program based on the amount they pay, up to a certain limit, and as I recall, the same amount is taken from the employees’ paychecks. This is paid to the government UI agency.

When an employee claims and is granted payments, the amount they and the employer put in is taken into account, determining both the amount of UI paid, and for how long it can be paid. And I know there have been some emergency extensions lately due to the insurance, from whence I know not how it is paid.

My question is, if an employee gets paid UI, does the employer end up paying more than they already did while the employee was employed? I didn’t think so, but I recently talked to an employer in California who was worried and thought that what happened was UI was tracked per company, and that the company’s former employees collected a lot of UI, that the company would need to pay more for this than they already had done in their employer matching when the former employees still worked for the company. Was he correct?

Sorry for the lengthy details, and thanks for any information.