Unknown software block of sound

As a volunteer managing two donated desktops at a seniors' residence, I was unable to activate the sound and the system tray icons are "X" ed out no matter what I tried:

I tested the sound with a live Linux Mint USB stick and the hardware is functioning with either speakers or headsets.

The computers are two identical HP Compaq dc7900 small form factor desktops, with Windows 7 Pro 64bit, SP1 with 4GB memory and everything is functioning well. So far, I checked the installed drivers and it reports that everything is working properly. But the Windows sound service cannot start, and the re-installation of the HP and SoundMax driver packages return the following message.

"The Windows Audio Service should work but could not start"

Also checked group rights, users rights, administrators rights but they are all undefined.

Hello ineuw mate have you tried thee yet?
SFC /SCANNOW Command - System File Checker
Disk Check < if necessary include the /f switch and if this does not make any improvement then just run this switch instead /r in the command line as per Option2
Run these in safe mode and the sfc often best run for two to three runs – you can leave out the /r switch in ckdsk if you feel it not necessary

if no fix then these

SUPERAntiSpyware | Remove Malware | Remove Spyware - AntiMalware, AntiSpyware, AntiAdware!
Malwarebytes | Malwarebytes Premium
AdwCleaner Download
ADW download from bleeping computer delete any rubbish found with the malware scans
(NB If you are running Kaspersky as the main AV then it might need disabling it when trying ADW because it does not like it at times)

You have already start the windows audio service, all right? Maybe you can check the BIOS and see if the integrated peripherals or onchip sound is disabled.

ICIT2LOL, thanks for the directions. Just managed to get access to the machines and that's why it took me so long to reply.

These computers were donated by a computer shop which checks them out, re-formats the drives and installs a licensed copy of Windows 7 Pro and MS Office 2010. The software you listed were already installed by me and performed all of the procedures when I took possession of the computers, but now I repeated them in safe mode and everything was OK. The antivirus software is Avast.

I also check and analyze the Events, and one error related to the sound is "The account specified for this service is different from the account specified for other services running in the same process."

Hi ineuw,

Try these commands, use Administrator account:

net localgroup Administrators /add networkservice
press enter then type:
net localgroup Administrators /add localservice
press enter then type:
exit
press enter and restart your computer

It's possible the computers' built-in speakers no longer work and/or motherboard's sound card's output no longer works; I had to use usb speakers or usb headphones to hear from my laptop.

Thanks for all the advice. The problem is definitely a permission issue according to the error numbers. Currently, I am replying from a cellphone, but will explain in detail the naming schemes used in the users and groups configuration of the system and this is the problem because this is a French version of Win 7, and I am an English speaker so I am slowed down a bit. All services necessary for the Windows sound is activated. As I posted originally, the sound works fine with a Linux mint live USB drive. Again, my thanks.

The audio service login was not selected. Then, I was searching for NT AUTHORITY\Local Service which doesn't exist on standalone machines. Only LOCAL SERVICE and that did the trick. Thanks again, it was an interesting discovery.