If I have a router that has several LANs on it, and each of these LAN is attached to a second router, do I need to use different HSRP groups for each LAN?

With this set up, each virtual gateway will be on a Layer 2 segment. And within a router, no interface will have multiple gateways.

So, For example:

Router 1:

F0/0:
ip address 192.168.1.2 255.255.255.0
standby ip 192.168.1.1
F2/0:
ip address 192.168.2.2 255.255.255.0
standby ip 192.168.2.1

Router 2:

F0/0:
ip address 192.168.1.3 255.255.255.0
standby ip 192.168.1.1
F2/0:
ip address 192.168.2.3 255.255.255.0
standby ip 192.168.2.1

Will this work, or do I need standby 1 ip 192.168.2.1 on the F2/0 interfaces? Since according to the RFC, the group number of the packet is in the HSRP multicast packets, my guess is that I don't need different groups, and that multiple groups are only needed when they are all on the same Layer 2 segment. However, I haven't been able to find this setup....