Let me edit this response in order to clarify what I am showing.
<p>

First, I will begin with an example: $t(x_1,x_2,x_3,x_4,x_5,x_6)
= ((((x_1x_2)x_3^{-1})x_3)x_4)x_4^{-1}$
is a group term. If you want to associate this term to a particular group, let it be the free group $F_6$ on $\{x_1,\ldots,x_6\}$. The term $t$ depends on $x_1$ and $x_2$.
The variables $x_3$ and $x_4$ occur,
but $t$ does not depend on them.
The variables $x_5$ and $x_6$ do not occur.
The permutations of indices that are symmetries of the original term $t$
are those from the full symmetric group $\textrm{Sym}(\{3,4,5,6\})$.
One may eliminate *fictitious variables*
by defining a term $s(x_1,x_2,x_3,x_4) = t(x_1,x_2,x_3,x_4,x_4,x_4)$.
The terms $s$ and $t$ are the same, all variables of $s$ occur in $s$, and $s$ does not depend
on its last two variables. The symmetries of $s$ are $\textrm{Sym}(\{3,4\})$. The term $r(x_1,x_2) = t(x_1,x_2,x_2,x_2,x_2,x_2)$ depends on all of its variables, which are the same variables that $t$ depends on, and $r$ acts the same way as $t$ with respect to those variables. The symmetry group of $r$ is trivial.

More generally, suppose that
$t(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_n)$
depends on $x_1$--$x_i$, does not depend on $x_{i+1}$--$x_j$ although these variables occur,
and the variables $x_{j+1}$--$x_n$ do not occur.
The term $r(x_1,\ldots,x_i):=t(x_1,\ldots,x_i,x_i,\ldots,x_i)$
will be a term that depends on all variables and will
have symmetry group equal to some $G\leq S_i$.
The term $s(x_1,\ldots,x_j):=t(x_1,\ldots,x_i,\ldots,x_j,x_j,\ldots,x_j)$
will be a term equal to $t$, where all variables occur, and will
have the symmetry group $G\times S_{j-i}$.
The original term $t(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_n)$
will have the symmetry group $G\times S_{n-i}$.<br>
The symmetry groups of terms are determined, up to symmetric group factors, by the symmetry groups
of just those terms that depend on all variables. Moreover, the symmetric group factors can be determined if you know which variables of the term occur and which variables the term depends on.
<p>

There are various questions that could be asked, such as<p>

<li> What are the possible symmetry groups of terms?
(Answer below: any concrete subgroup of $S_n$
is the symmetry group of an $n$-ary term that depends on all variables.)<br>
<li> What are the possible classes of permutation groups
on finite sets which are the symmetry groups of those
terms of an algebra (i) which depend on all variables?
(ii) in which all variables occur? (iii) are arbitrary?
The question asked is (ii), but I will answer (i) here.<p>

I am going to explain why, if $\mathcal K$ is any class of groups
whose members are subgroups of finite symmetric groups and $\mathcal K$ contains $S_1$, then there is an algebra $A$ such that the symmetry groups $G(t)$ of terms $t$ of $A$ that depend on all variables are exactly the groups in $\mathcal K$. (More precisely, if $G\in {\mathcal K}$ is a subgroup of $S_n$, then we will realize $G$ and all of its conjugates in $S_n$ as symmetry groups of terms of $A$ which depend on all variables.) We need to include $S_1$ in $\mathcal K$ because $t(x)=x$ has symmetry group $S_1$.

<p>
 
For a given $G\in\mathcal K$,
where $G\leq S_n$,
define an $n$-ary operation $f_G(x_1,\ldots,x_n)$ on the set
$\{a, b\}\cup \mathbb N^+
= \{a, b, 1, 2, 3, \ldots\}$ as follows:
$$
f_G(\bar{u}) = \begin{cases} a & \textrm{if $\bar{u} = 
(\sigma(1),\ldots,\sigma(n))$ for some $\sigma\in G$;}\\
b & \textrm{otherwise.}
 \end{cases}
$$
Let $A$ be the algebra on $\{a,b\}\cup \mathbb N^+$
whose operations are all operations
of the form $f_G$, $G\in\mathcal K$.
<p>

I claim that the following are true.<br>
* Each $f_G$ depends on all of its variables.<br>
* The symmetry group of $f_G(x_1,\ldots,x_n)$ is $G\;(\leq S_n)$.<br> 
* any term operation of $A$ that depends on all of its
variables is obtained from one of the $f_G$'s by
permuting the variables. Hence, its symmetry
group is conjugate to $G$ in $S_n$.<p>


If $f_G(x_1,\ldots,x_n)$ is defined as above, and we identify two variables: $f_G(\underline{x_2},\underline{x_2},x_3,x_4,\ldots,x_n)$, then this operation is constant and its variable-set is $\{x_2,\ldots,x_n\}$.
Its symmetry group in Noah's sense is $S_{n-1}$. Similarly, the composite $f_G(f_G(x_1,\ldots,x_n),y_2,\ldots,y_n)$ is constant and its variable-set is $\{x_1,\ldots,x_n,y_2,\ldots,y_n\}$, so its symmetry group is $S_{2n-1}$. Using tricks like this one can obtain every finite symmetric group as a symmetry group of a term of $A$, as long as $A$ has at least one operation $f_G$ of arity greater than $1$. 
The isomorphism types of Noah-symmetry groups turn out to be 
exactly those in $\mathcal K\cup \{S_n\;|\;n=1, 2, \ldots\}$.