Indeed Gil Amelio bought NeXT which brought Steve back to Apple in December 1996. How he took over, however, is a different story.

As you can see in this 1997 NYT article: http://www.nytimes.com/1997/07/10/business/apple-computer-ousts-chief-in-response-to-poor-results.html
Amelio was ousted after a board coup in July 1997. The article mentions that the tension leading up to Amelio's departure was in part brought up by Larry Ellison, CEO of Oracle and one of Steve's best friends - who spread rumors he was about to buy Apple. They also mention Mike Markkula (the first investor in Apple, back in 1977) as the one who informed Gil Amelio of his demise.
However they don't talk of who I think was the most important player in bringing back Steve, Jerry York, former CFO of Chrysler (I read this in Alan Deutschman's The Second Coming of Steve Jobs).

After he took over as 'interim CEO', Steve changed a number of board members. The only two members from the old board left were Ed Woolard and Gareth Chang. The new members: Bill Campbell... and of course Jerry York, Larry Ellison, and Steve himself.