My GH is also a 2004 however it's not the fuel solenoid. It's getting gas to the carb just not firing. After writing the initial post I had determined it was an electrical problem. I am an electrical engineer and I know from past experience it's good to have a starting point for testing. I wrote GH Customer Service an email explaining the problem. Dick from GH called me 4 hours later. His first suggestion was to check the fuses. Second, check the connections at the fuse block. 3rd, check relay "A". 4th, he gave me the values to check the coil. With this info I am confident in finding the problem.
Thanks bob & Degolyerent for your help. This is a great forum and I am glad I found it. I have a lot of yard maint equip and I'am sure to return.
I will post when the problem is found and repaired.