Harry Potter question - How did Kreacher betray Sirius to Voldemort?

When Sirius ordered Kreatcher to "Get out" (of the kitchen), Kreatcher took that as an order to get out of the house so he went to the family member that he considered to be most in keeping with the Black family 'values', this being Bellatrix Lestrange. The magical enchantments that bind house-elves to their masters prevented him from betraying proper 'secrets', but he did tell her of the great love and attachment that had built up between Harry and Sirius. Bellatrix then took this information to Voldemort, who was already aware of Harry's saving people 'thing'. He then used this to plant the dream that Sirius had been captured and was being tortured in the Ministry. When Harry attempted to check whether Sirius was still at Grimmauld Place or not, via floo powder in Umbridge's office fireplace, Kreatcher then lied to Harry and told him that Siruis had gone - on Bellatrix's orders - this really being the actual act of betrayal. Because of this Harry went to the Ministry thinking he was saving Sirius, when in fact this made Sirius go there to save Harry leading to Sirius's death at the hands of Bellatrix.