Good luck to the search committee of General Motors in finding a new chief executive.

The abrupt departure of Fritz Henderson from the job this afternoon under pressure from Ed Whitacre, GM’s chairman (and temporary replacement as chief executive) leaves the company looking for someone to take his place.

The first question any candidate is going to ask himself or herself, however, is: what will it be like working under an impatient, strong-willed and sometimes abrasive chairman who made little secret of his lack of belief in Mr Henderson?

The job of non-executive chairman has never really caught on in the US, although it is commonplace in the UK, and Mr Whitacre, the former chairman and chief executive of AT&T, has not given the impression since arriving at GM that he believes in the notion.

Mr Henderson’s position was undermined by the GM board’s decision to over-rule him on the sale of Opel, its European subsidiary, last month and Mr Whitacre’s open disagreement on issues such as the timing of GM’s initial public offering and the speed of internal change.

There was no doubt plenty in GM’s slow-moving and inwardly-focused culture to concern Mr Whitacre, but it will not make it easy to find a replacement.

The model is probably Ford, which managed to attract Alan Mulally from Boeing in 2006, although Bill Ford remained as executive chairman. Mr Mulally judged, apparently correctly, that Mr Ford would given him freedom of manoevre at the company.

That parallel occurred to Steve Rattner, the former head of the US government’s auto task force, when he was considering whether Mr Henderson should take over from Rick Wagoner as GM’s chief executive in March.

As Mr Rattner wrote in Fortune magazine in October:

“The question for us was whether GM would be better off with Fritz or with an outsider, as Ford had done in bringing in [Mr Mulally]. While nervous about whether Fritz could bring the change GM desperately needed, I was considerably more nervous about the likelihood of recruiting a thoroughbred CEO in the midst of the turmoil.”

GM now has its work cut out to persuade candidates that there is a job for a thoroughbred CEO under a chairman who likes to run things.

Update: Last March, I gave Mr Henderson a 50-50 chance of surviving at the helm of GM on the grounds that he was “a tough and talented executive but indubitably an insider” who might have “damaged his long-term chances by pinning his colours so firmly to Mr Wagoner’s mast.”