You can run 'df' to list the mounted filesystems to determine where home is. Basically, if you don't see '/home' listed next to anything under the 'Mounted On' column, then it's under root.
With an LVM setup remember that you logical volumes which contain the filesystems. You'll need to change the sizes of things in the correct order (shrink the filesystem before reducing the logical volume, extend the logical volume before growing the filesystem).
Here's the gist of what you need to do:
1. Unmount the /srv filesystem.
2. Reduce the logical volume with the lvreduce command (use the ---resizefs command to shrink the filesystem). You can mount the /srv filesystem again after doing so.
https://docs.redhat.com/en/documentation/red_hat_enterprise_linux/6/html/logical_volume_manager_administration/lv_reduce
3. Extend the root volume with the lvextend command.
https://docs.redhat.com/en/documentation/red_hat_enterprise_linux/6/html/logical_volume_manager_administration/lv_extend
4. Extend the root filesystem with resize2fs.
https://askubuntu.com/questions/24027/how-can-i-resize-an-ext-root-partition-at-runtime
Alternatively, instead of allocating the space to the root volume you could create a new logical volume for the home partition. To accomplish this, you still need to perform steps 1 and 2 above to free up the space. Then you can do the following:
1. Use lvcreate to create a new logical volume: https://commandmasters.com/commands/lvcreate-linux/
2. Create a filesystem on the logical volume:
https://www.golinuxcloud.com/create-filesystem-partition-linux-lvm/
3. Mount the filesystem somewhere other than home.
4. Copy/move the files contained within the /home directory into the new filesystem. Make sure to preserve permissions.
5. Unmount the filesystem from its temporary mountpoint, mount it at /home, modify your /etc/fstab file to add the /home mountpoint for the new filesystem.
Probably should say to be careful when doing this. Mistakes may result in data loss or situations where the system won't boot which may be challenging to troubleshoot.