Aims. We present an experimental investigation of the exothermic reactions of NH⁺, NH₂⁺, and NH₃⁺ ions with H₂ at temperatures relevant for interstellar clouds. Methods. The reactions were studied using a variable-temperature 22-pole radio frequency ion trap instrument. Results. The temperature dependences of rate coefficients of these reactions have been obtained at temperatures from 15 up to 300 K. The reaction of NH⁺ with H₂ has two channels, which lead to NH₂⁺ ( ∼ 97%) and H⁺₃ ( ∼ 3%) with nearly constant reaction rate coefficients (k^a_NH⁺(17 K) = 1.0 × 10⁻⁹ cm³ s⁻¹ and k^b_NH⁺(17 K) = 4.0 × 10⁻¹¹ cm³ s⁻¹, respectively). The reaction of NH₂⁺ with H₂ produces only NH₃⁺ ions. The measured rate coefficient monotonically decreases with increasing temperature from k_NH2⁺(17 K) = 6 × 10⁻¹⁰ cm³ s⁻¹ to k_NH2⁺(300 K) = 2 × 10⁻¹⁰ cm³ s⁻¹. The measured rate coefficient of the reaction of NH₃⁺ with H₂, producing NH⁺₄, increases with decreasing temperature from 80 K down to 15 K, confirming that the reaction proceeds by tunnelling through a potential barrier.