Southgate, initially appointed on an interim basis, was seen as the squeaky clean role model the Football Association needed after Sam Allardyce was brought down by a newspaper sting.
Sam Allardyce has spearheaded a number of unlikely, late-season escape acts in the Premier League down the years.
Sam Allardyce has been tasked with an overhaul of the system at the club, having first been contacted by the Toffees two years ago prior to Walsh's appointment.
Sam Allardyce was sacked as manager of Premier League side Everton despite guiding them away from the relegation.
Sigurdsson's stunning goal against his former club maintained Everton's unbeaten record under Sam Allardyce with a 3-1 win.
Hired on Thursday, Allardyce will not have been impressed by Everton's poor first-half display but Sigurdsson struck in the 47th minute from Calvert-Lewin's flick.
David Unsworth has been in temporary charge since Koeman was sacked on October 23 and had made clear his desire to take on the job permanently.