Phew, that one took a while to finally figure out.

Here’s how I could succeed in doing this:

• Shutdown the Virtual Machine
• Increase the size of the HD: VBoxManage.exe modifyhd <absolute path to file> --resize <new total size in MB>
• Download the gparted iso, mount it in your VM, start it from this iso
• Increase the partition in gparted (you might need to adjust the “outer” partition as well)
• Reboot and remove the gparted iso
• Extend the logical volume: sudo lvextend -L30G /dev/mapper/your-partition-name-root
• Resize the filesystem: sudo resize2fs /dev/your-partition-name/root
• Check that everything worked: sudo lsblk