extend (one's) sympathy
To share one's condolences. I'm so sorry to hear about your sister's passing. Please extend my sympathy to your mother as well. Don't worry, I extended your sympathy in my card too.
extend one's sympathy (to someone)
to express sympathy to someone. (A very polite and formal way to tell someone that you are sorry about a misfortune.) Please permit me to extend my sympathy to you and your children. I'm very sorry to hear of the death of your husband. Let's extend our sympathy to Bill Jones, who is in the hospital with a broken leg. We should send him some flowers.