using time capsule to extend a network

Yes, if you change the settings on the Time Capsule to have it produce a wireless network that uses the exact same wireless network name, same wireless security and and same wireless password as your existing network. Then, make sure that the Time Capsule is set up to operate in Bridge Mode.

To do this....

Open AirPort Utility on your Mac (Finder > Applications > Utilities > AirPort Utility)

Click on the picture of the Time Capsule

Click Edit in the smaller window that appears

Click the Wireless tab at the top of the next window

Edit the Wireless Network Name to match the name of your existing network

Make sure that the setting for Wireless Security is WPA2 Personal

Edit the Wireless Password to match the password of your existing network

Edit Verify Password to match the password again

Click the Network tab at the top of the window

Make sure that the Router Mode is set to "Off (Bridge Mode)"

Click Update at the lower right of the window and give the Time Capsule a full minute to restart

No, the Time Capsule cannot wirelessly extend the network provided by a third party router. It can only "extend" the network using wireless if your "main" router is also another Apple router.

The Time Capsule must be connected to your existing third party router with an Ethernet connection at all times.