This paper presents a novel approach to autonomous extinguishing of indoor fires inside a building by a Micro-scale Unmanned Aerial Vehicle (MAV). In particular, controlling and estimating the MAV state, detection of a building entrance, multi-modal MAV localization during the outdoorindoor transition, interior motion planning and exploration, fire detection and position estimation, and fire extinguishing are discussed. The performance of these elements, as well as of the entire integrated system, are evaluated in simulations and field tests in various demanding real-world conditions. The system presented here is part of a complex multi-MAV solution that won the Mohamed Bin Zayed International Robotics Challenge 2020 (MBZIRC 2020) competition, and is being used as the core of a fire-fighting Unmanned Aerial System (UAS) industrial platform under development. A video attachment to this paper is available at the website http://mrs.felk.cvut.cz/2020firechallenge-insidefires.