Injective mappings of words that do not propagate distortions of letter omission type

Let $A^*$ be the set of all words of finite length in an alphabet $A$. A complete description of all injective maps of the set $\Omega^*$ into the set $\Omega_1^*$ that do not multiply symbol skip errors is given. We assume that the alphabets $\Omega$ and $\Omega_1$ are finite.