Confusion regarding computing Multiplicative Inverse Modulo P?
Indeed, theoretically $$X'\equiv C^{a^{-1}}\equiv A^{x*a^{-1}}\equiv g^{x}=X\pmod p$$

should hold, given $gcd(a,p-1)=1$.

However there are two reasons why you won't get this equation to hold.


You calculated $a^{-1}$ using the wrong modulus. As it is in the exponent you need to calculate $a^{-1}\bmod q$ with $q$ being the group's order instead of $a^{-1}\bmod p$. In your example $q=p-1=10$.
The value you have for $a$ makes it impossible to invert. For $a^{-1}\bmod q$ to exist $gcd(a,q)=1$ must hold, but $gcd(6,10)=2\neq1$ so there's no $a^{-1}\bmod q$ for your specific choices of $a$ and $q$.