When you have group coverage, you have the option to remain on the group plan if you choose. COBRA is the option some former employees choose so their is no break in coverage and no gap in medical services. Usually, COBRA is allowed for 18 months after you stop working for your company. In some cases, COBRA can be extended if you have a qualifying event for an additional 18 months. Important to note, COBRA is much more expensive than regular coverage through your employer because during COBRA you have to pay the entire bill.
Before, your employer contributed a heavy portion of the monthly premium so that is why it can be quite expensive. Also, companies can charge an additional 2% administration fee on top of the monthly premium. At the end, it might benefit the former employee to shop for individual health insurance as that might be the cheaper option for medical coverage.