‘A while’ vs. ‘awhile’

According to my spot research, I notice that yes, both are perfectly correct as such. However, they are not strictly interchangeable.

"Awhile" is a single adverb. "For a while," that is identical in meaning, is a construct called an "adverbial phrase," or a phrase used as if it was a single adverb.

Be careful here. The phrase "in a while" is also an adverbial phrase, but it is not identical in meaning to "a while" or "for a while."

They are not the same. In fact, I don't know if you can ever interchange them. You cannot, for example change "let's go out for a while" with "let's go out for awhile." Nor can you change "this job will take a while" with "this job will take awhile". "Awhile" is interchangeable with "for a while".

The CMS addresses a similar question
http://ucp.uchicago.edu:2001/search97cgi/s97_cgi?action=View&VdkVgwKey=%2E%2E%2F%2E%2E%2FWWW%2Dbooks%2Fmain%2FMisc%2FChicago%2Fcmosfaq%2Fcmosfaq%2EUsage%2Ehtml&DocOffset=1&DocsFound=1&QueryZip=awhile&Collection=C90&SearchUrl=http%3A%2F%2Fucp%2Euchicago%2Eedu%3A2001%2Fsearch97cgi%2Fs97%5Fcgi%3Faction%3DFILTERSEARCH%26QueryZip%3Dawhile%26Filter%3Dcmosfaq%252Ehts%26ResultTemplate%3DCMOSfaqresults%252Ehts%26QueryText%3Dawhile%26Collection%3DC90%26ResultStart%3D1%26ResultCount%3D10&

Directly from Dictionary.com:

"Awhile, an adverb, is never preceded by a preposition such as for, but the two-word form a while may be preceded by a preposition. In writing, each of the following is acceptable: stay awhile; stay for a while; stay a while (but not stay for awhile)."

these information help a lot, thanks!