On which of Lucille Ball's four U.S. TV sitcom shows did she play a character whose last name was Carmichael?

Regarding Lucille Ball's four U.S. TV sitcom shows, the one where she plays a character whose last name is Carmichael is "The Lucy Show" (1962 – 1968). This show began with Lucille Ball as Lucy Carmichael, a widow with two children, Chris (Candy Moore) and Jerry (Jimmy Garrett), living in the fictional town of Danfield, New York. She is sharing her home with divorced friend Vivian Bagley (Vance) and her son, Sherman (Ralph Hart).

"The Lucy Show" is an American sitcom that aired on the CBS Network. It was Lucille Ball's follow-up to "I Love Lucy" (1951 – 1957). A significant change in cast and premise took place on "The Lucy Show" for the fourth season (1965–1966). The show was divided into two distinct eras; aside from Ball, only Gale Gordon, who joined the program for its second season, remained. For the first three seasons, Vivian Vance was the co-star.

The producers at CBS took their time before they decided on the title "The Lucy Show". Lucy Ball won consecutive Emmy Awards as Outstanding Lead Actress in a Comedy Series for the series' final two seasons, 1966–67 and 1967–68.