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The 24-year-old moved to Harriers from Walsall last December following Stuart Watkiss appointment at Aggborough, and scored three times in 14 appearances before the end of the season.
Alexander moved for the Birmingham-born striker, who has also had loans spells at Barnsley and Exeter after losing two of his own strikers in Yeo and Gary Taylor Fletcher. However news that Birch has scored less goals in his career than Yeo did last season may dismay City fans.