I'm not sure I understand the difference between going under and going into admin.
Your equivalent of administration is a chapter 11 bankruptcy. Administration is a formal insolvency process which was brought in back in 2003 to try to rescue businesses by allowing an administrator to go in and manage things whilst marketing the business for sale to the highest bidder. The added benefit is the administration creates a moratorium on legal action, so no one can sue or enforce their debts against the company whilst the administrator is in office.
The business and assets (which includes the team, coaching staff, etc) are then sold to a new company, with the debts left in the old company (and the administrator will use the sum paid by NewCo to pay off some of the debts in OldCo).
By contrast, our other main formal insolvency process in Eng & Wales for corporate entities is a liquidation (also known as "winding up" or, colloquially, going "bust"), which is the death of the business and of the company. So, if it's a football club, it no longer exists if the company goes into liquidation. Liquidation is the only option if there is no way of rescuing the business. This is what happened to Bury.
To avoid encouraging owners to allow clubs to rack up huge debts, and then just putting them into administration and buying the business back using a new company, the PL and football league apply heavy points deductions if a club goes into administration. Recent examples include Derby County.
As you may have guessed, I work in insolvency!