A Mac server can do it via netboot, but it is an incredibly complex process for one machine, and who wants to buy server & use it once
Deploy Studio can do the same, but again it is complex, you would need to make a netboot image of the working Mac (and hope it can support both machines) & then you can boot across the network & run the installer.
Can't you restore the install.dmg to a USB flash drive or to another disk or partition? I think you use the Restore tab in Disk Utility to copy the image onto the other drive. I think all Macs support USB booting nowadays.
Otherwise if you can get the Lion installer onto the Mac it can do an install from the internet, but be prepared to wait hours for it.