The group of connected components of the subgroup of $Aut(X)$ fixing $H^{11}(X)$ is finite. This follows from the main result of

Polarized Varieties, Fields of Moduli and Generalized Kummer Varieties of Polarized Abelian Varieties, T. Matsusaka, American Journal of Mathematics, Vol. 80, No. 1 (Jan., 1958), pp. 45-82

I quote from the mathscinet review:

The main result (section 3) is as follows. Let $U$ be a projectively embeddable non-singular variety; then there is a maximal connected algebraic group $G$ of automorphisms of $U$. Moreover, $G$ is of finite index in the group $G′$ of those automorphisms of $U$ which, for a given projective embedding of $U$, carry a hyperplane section $H$ into a cycle numerically equivalent to $H$.

Since the group of numerical equivalence classes of divisors embeds in $H^{11}$, the group of automorphisms preserving $H^{11}$ is a subgroup of the group $G'$ above. It is clearly a closed algebraic subgroup, so it likewise has finitely many components.

I must admit, on a quick read, I also endorse the last sentences of the review:

Unfortunately, the author's style, added to the intrinsic difficulties of the subject, makes it exceedingly hard to check the accuracy of many technical details which seem essential for the validity of the proofs; this is all the more to be regretted, since his results are so valuable and important. It is to be hoped that he, or someone else, will some day give a completely lucid and completely convincing exposition of these topics.

The group of connected components of the subgroup of $Aut(X)$ fixing $H^{11}(X)$ is finite. This follows from the main result of

Polarized Varieties, Fields of Moduli and Generalized Kummer Varieties of Polarized Abelian Varieties, T. Matsusaka, American Journal of Mathematics, Vol. 80, No. 1 (Jan., 1958), pp. 45-82

I quote from the mathscinet review:

The main result (section 3) is as follows. Let $U$ be a projectively embeddable non-singular variety; then there is a maximal connected algebraic group $G$ of automorphisms of $U$. Moreover, $G$ is of finite index in the group $G′$ of those automorphisms of $U$ which, for a given projective embedding of $U$, carry a hyperplane section $H$ into a cycle numerically equivalent to $H$.

Since the group of numerical equivalence classes of divisors embeds in $H^{11}$, the group of automorphisms preserving $H^{11}$ is a subgroup of the group $G'$ above. It is clearly a closed algebraic subgroup, so it likewise has finitely many components.

I must admit, on a quick read, I also endorse the last sentences of the review:

Unfortunately, the author's style, added to the intrinsic difficulties of the subject, makes it exceedingly hard to check the accuracy of many technical details which seem essential for the validity of the proofs; this is all the more to be regretted, since his results are so valuable and important. It is to be hoped that he, or someone else, will some day give a completely lucid and completely convincing exposition of these topics.

The answer is no. This is the main result of

Polarized Varieties, Fields of Moduli and Generalized Kummer Varieties of Polarized Abelian Varieties, T. Matsusaka, American Journal of Mathematics, Vol. 80, No. 1 (Jan., 1958), pp. 45-82

I quote from the mathscinet review:

The main result (section 3) is as follows. Let $U$ be a projectively embeddable non-singular variety; then there is a maximal connected algebraic group $G$ of automorphisms of $U$. Moreover, $G$ is of finite index in the group $G′$ of those automorphisms of $U$ which, for a given projective embedding of $U$, carry a hyperplane section $H$ into a cycle numerically equivalent to $H$.

Since the group of numerical equivalence classes of divisors embeds in $H^{11}$, the group of automorphisms preserving $H^{11}$ is a subgroup of the group $G'$ above. It is clearly a closed algebraic subgroup, so it likewise has finitely many components.

I must admit, on a quick read, I also endorse the last sentences of the review:

Unfortunately, the author's style, added to the intrinsic difficulties of the subject, makes it exceedingly hard to check the accuracy of many technical details which seem essential for the validity of the proofs; this is all the more to be regretted, since his results are so valuable and important. It is to be hoped that he, or someone else, will some day give a completely lucid and completely convincing exposition of these topics.

The group of connected components of the subgroup of $Aut(X)$ fixing $H^{11}(X)$ is finite. This follows from the main result of

Polarized Varieties, Fields of Moduli and Generalized Kummer Varieties of Polarized Abelian Varieties, T. Matsusaka, American Journal of Mathematics, Vol. 80, No. 1 (Jan., 1958), pp. 45-82

I quote from the mathscinet review:

The main result (section 3) is as follows. Let $U$ be a projectively embeddable non-singular variety; then there is a maximal connected algebraic group $G$ of automorphisms of $U$. Moreover, $G$ is of finite index in the group $G′$ of those automorphisms of $U$ which, for a given projective embedding of $U$, carry a hyperplane section $H$ into a cycle numerically equivalent to $H$.

Since the group of numerical equivalence classes of divisors embeds in $H^{11}$, the group of automorphisms preserving $H^{11}$ is a subgroup of the group $G'$ above. It is clearly a closed algebraic subgroup, so it likewise has finitely many components.

I must admit, on a quick read, I also endorse the last sentences of the review:

Unfortunately, the author's style, added to the intrinsic difficulties of the subject, makes it exceedingly hard to check the accuracy of many technical details which seem essential for the validity of the proofs; this is all the more to be regretted, since his results are so valuable and important. It is to be hoped that he, or someone else, will some day give a completely lucid and completely convincing exposition of these topics.