Algebras simple with respect to a Sweedler's algebra action

Algebras simple with respect to an action of Sweedler's algebra $H_4$ deliver the easiest example of $H$-module algebras that are $H$-simple but not necessarily semisimple. We describe finite dimensional $H_4$-simple algebras and prove the analog of Amitsur's conjecture for codimensions of their polynomial $H_4$-identities. In particular, we show that the Hopf PI-exponent of an $H_4$-simple algebra $A$ over an algebraically closed field of characteristic $0$ equals $\dim A$. The groups of automorphisms preserving the structure of an $H_4$-module algebra are studied as well.